Writing Proofs

Direct Proof. Contrapositive. Proof by contradiction. Mathematical induction. Strong Form of Induction. Read more... 1718 words,🕔9 minutes read, May 14, 2022.

The Natural Numbers. The Binomial Theorem.

The natural numbers. The binomial theorem. Well-ordering property. Pascal's Triangle. Read more... 2374 words,🕔12 minutes read, May 14, 2022.

Integers II. Euler's totient.

φ(1) = 1, φ(n) = n° a, a≤n, (a, n) = 1. If p prime, φ(p) = p-1. φ is multiplicative, (a, b) = 1 ⇒ φ(ab) = φ(a)φ(b). The Method of Repeated Squares. Read more... 693 words,🕔4 minutes read, Jan 14, 2022.

Integers. Well Ordering Principle. Division & Euclidean Algorithm.

Well Ordering Principle. Divisibility Basics. Division Algorithm. Greatest Common Divisor. Least Common Multiple. Fundamental Theorem of Arithmetic. Euclidean Algorithm. Read more... 2616 words,🕔13 minutes read, Jan 14, 2022.

Sets. Algebra of sets

Sets. Algebra of sets. Distributive Laws. Double inclusion. De Morgan's laws. Truth tables. Cardinality. Properties of Set Algebra. Read more... 3117 words,🕔15 minutes read, May 14, 2022.

Relations

Cartesian Product. Relations. Definition, examples, and properties. Equivalence relations. Partitions. Equivalence Classes and Partitions. Read more... 3116 words,🕔15 minutes read, Jul 12, 2022.

Functions or Mappings

Definition, type, and properties. Counterexamples. Composition of functions. Types of functions. Injective, surjective, and bijective functions. Pigeonhole principle. Read more... 1580 words,🕔8 minutes read, May 14, 2022.

Functions or Mappings II

Composition of functions. Properties of Composition of Functions. Inverse functions. Cardinality of sets. Cantor's Theorem. A function is bijective if and only if has an inverse. Read more... 2461 words,🕔12 minutes read, May 14, 2022.

Groups. Binary Operations.

Binary Operations. Groups. Definition and examples. Read more... 2158 words,🕔11 minutes read, May 15, 2022.

Groups II. Properties.

Order of Groups & elements. Group properties. Uniqueness of identity and inverses. Cancellation property. Left inverse for all is right inverse. The shoe and sock principle. ax = b and xa = b have unique solutions in G. The laws of exponents for groups. Read more... 1751 words,🕔9 minutes read, May 14, 2022.

Groups III. Symmetries. Group Actions.

(ab)² = a²b² ⇒ ab = ba. If every non-identity element of G has order 2 ⇒ G Abelian. Cayley Tables. Indeed group theory is the mathematical language of symmetry. Read more... 1705 words,🕔9 minutes read, May 14, 2022.

Subgroups.

Subgroups. Subgroup tests (a·b⁻¹∈H). Finite subset are subgroups ↭ closed. H ≤ K ≤ G ⇒ H ≤ G. H ≤ G, L ≤ G, ⇒ H∩L ≤ G. The cyclic subgroup generated by a ⟨a⟩. Read more... 2093 words,🕔10 minutes read, May 18, 2022.

Center, Centralizers, & Normalizers.

Centralizers are subgroups. The center of a group is the set of elements that commute with all the elements of G. Z(G) ≤ G. Read more... 1077 words,🕔6 minutes read, May 17, 2022.

Conjugate of subgroups. Product of group subsets.

G Abelian, H₁ is conjugate H₂ ↭ H₁ = H₂. G Abelian, g₁ is conjugate g₂ ↭ g₁ = g₂. G group, g₁ is a conjugate of g₂ ⇒ |g₁| = |g₂|. H ≤ G ⇒ H⁻¹ = H. HK ≤ G ↭ HK = KH. |HK| = |H||K|/|H∩K|. Read more... 1452 words,🕔7 minutes read, May 17, 2022.

Cyclic Groups

Cyclic Groups. If |a| = ∞, a^i=a^j ↭ i = j. Otherwise, ⟨a⟩ = {1, a, a²,···, a^(n-1)} distinct, a^i=a^j ↭ n | (i -j). |a| = |⟨a⟩| = {1, a, a²,···, a^(n-1)}. Let a ∈ G, |a|< ∞, a^m = e ↭ |a| | m Read more... 2038 words,🕔10 minutes read, May 14, 2022.

Cyclic Groups II. Fundamental Theorem.

Cyclic group ⇒ Abelian. Fundamental Theorem of Cyclic Groups. Subgroups of cyclic groups are cyclic. If ⟨a⟩ = n, H ≤ ⟨a⟩ ⇒ |H| | n. For each positive divisor k of n, ∃! H ≤ ⟨a⟩, |H| = k, namely ⟨a^(n/k)⟩. G = ⟨a⟩ ⇒ G ≋ ℤ (|G|< ∞) or G ≋ ℤn. ⟨a^k⟩ = ⟨a^gcd(n,k)⟩. Read more... 2797 words,🕔14 minutes read, May 14, 2022.

Symmetric Groups

Permutation Groups. Properties of Permutations. |Sx| = |X|! Every permutation can be written as a product of disjoint cycles. Disjoint Cycles Commute. The order of a permutation = lcm(lengths of the disjoint cycles). Read more... 2486 words,🕔12 minutes read, May 14, 2022.

Symmetric Groups II

Every permutation is a product of 2-cycles. A permutation cannot be written as a product of both an odd and an even number of 2-cycles. The Alternating Group An ≤ Sn. |An| = n/2 . Read more... 1092 words,🕔6 minutes read, May 14, 2022.

The dihedral group

s reflection, srᵏs⁻¹ = r⁻ᵏ. Dₙ = {e, r, r²,..., rⁿ⁻¹, s, rs, r²s,..., rⁿ⁻¹s} = ⟨r, s | rⁿ = s² = e, rs = srⁿ⁻¹⟩. |Dₙ| = 2n. rᵏs = srⁿ⁻ᵏ, 1 ≤ k ≤ n-1. Read more... 2298 words,🕔11 minutes read, May 14, 2022.

Isomorphisms. Cayley's Theorem.

Two groups are isomorphic if there exist a bijective homomorphism, i.e., a one-to-one and onto map such that the group operation is preserved. Cayley's Theorem. Every group is isomorphic to a subgroup of a symmetric group. Read more... 2103 words,🕔10 minutes read, May 23, 2022.

Isomorphisms II. Properties.

Isomorphism carries the identity & generator (G = ⟨a⟩ ↭ G' = ⟨Φ(a)⟩), Φ(aⁿ) = Φ(a)ⁿ, preserves commutativity and order (|a| = |Φ(a)|), corresponding equations have the same number of solutions. Φ G → G' isomorphism ⇒ Φ⁻¹ isomorphism, G Abelian/Cyclic ↭ G' Abelian/Cyclic. Φ isomorphism, H ≤ G ⇒ Φ(H) ≤ G' Read more... 1546 words,🕔8 minutes read, May 14, 2022.

Automorphisms

Inner Automorphisms. Aut(G) and Inn(G) are both groups. Inn(G) ◁ Aut(G). Aut(ℤn) ≋ Un and |Aut(ℤn)| = Φ(n) where Φ is Euler's toilet function. Read more... 1767 words,🕔9 minutes read, May 12, 2022.

Partially Ordered Sets. Hasse Diagrams.

A partial order set is a subset of the Cartesian product ⊆ X x X such that is reflexive, antisymmetric, and transitive. Hasse Diagrams. Read more... 982 words,🕔5 minutes read, May 14, 2022.

Roots of Unity.

z = x + iy = r(cosθ + isinθ). Euler's Formula, z = r(cosθ + i·sinθ) = re^(iθ). ξₙ = e^(2π/n) is a primitive nth root of unity. Zₙ = the set of all nth roots of unity = ⟨ξ⟩ ≤ ℂ*. The circle group S¹ = {z ∈ ℂ* | |z| = 1} ≤ C* which contains Zₙ. Read more... 1558 words,🕔8 minutes read, May 14, 2022.

Cosets

Definition, examples, and properties of cosets. a ∈ aH. aH = H ↭ a∈H. aH = bH ↭ a ∈ bH. aH = bH or aH ∩ bH = ∅. aH = bH ↭ a⁻¹b ∈ H. ∀g∈G, |H| = |gH|. ∀a, b∈ G, |aH| = |bH|. aH = Ha ↭ H = aHa⁻¹. aH ≤ G ↭ a ∈ H. Read more... 2208 words,🕔11 minutes read, May 14, 2022.

Cosets II

H ≤ G ⇒ g₁H = g₂H ↭ Hg₁⁻¹ = Hg₂⁻¹↭ g₁H ⊆ g₂H ↭ g₁ ∈ g₂H ↭ g₁⁻¹g₂ ∈ H. The quotient set, G/H = {aG | a ∈ G}. The index of H in G, [G:H] = |G/H|. Let H ≤ G ⇒ [G/H] = [H\G] Read more... 1132 words,🕔6 minutes read, May 14, 2022.

Lagrange's Theorem

Lagrange's Theorem. |H| | |G|. |a| divides |G| Classification of Groups of Order 2p, G ≋ ℤ2p or Dp. Read more... 2521 words,🕔12 minutes read, May 14, 2022.

Orbits and Stabilizers

Orbit-Stabilizer Theorem. G a finite group of permutations of a set S, ∀i ∈ S, |G| = |orb(i)| |stab(i)| Read more... 2843 words,🕔14 minutes read, May 14, 2022.

Direct Products

A group of order 4 is isomorphic to ℤ₄ or ℤ₂ ⊕ ℤ₂. Any cyclic group of even order has exactly one element of order 2. The direct product of Abelian groups is also Abelian. |(g₁, g₂,..., gₙ)| = lcm(|g₁|, |g₂|,...,|gₙ|) Read more... 3174 words,🕔15 minutes read, May 14, 2022.

Direct Products II

If r | m and s | n, ℤm ⊕ ℤn has a subgroup isomorphic to ℤr ⊕ ℤs. Criterion for the direct product to be cyclic, G ⊕ H is cyclic ↭ (|G|, |H|) = 1. m = n₁·n₂···nₖ, ℤm ≋ ℤn₁ ⊕ ℤn₂ ⊕···⊕ ℤnₖ ↭ (ni, nj) = 1∀ i,j. m = n₁·n₂···nₖ, ∀i, j, i ≠ j, (ni, nj) = 1) ⇒ U(m) ≈ U(n₁)⊕U(n₂)⊕···⊕U(nₖ). Every group U(n) is isomorphic to the external direct product of cyclic groups. Read more... 2448 words,🕔12 minutes read, May 14, 2022.

The Integers modulo n. The Group of Units. Fermat’s Theorem.

The Integers modulo n. An integer has a multiplicative inverse modulo n ↭ (a, n) = 1. Uₙ = {a ∈ Zₙ | gcd(a, n)=1}, |Uₙ| = φ(n). The subgroup generated by an element k, k | n, Uₖ(n) = {x ∈ U(n) | x mod k = 1} ≤ U(n). m = n₁·n₂···nₖ, ∀i, j, i ≠ j, (ni, nj) = 1) ⇒ U(m) ≈ U(n₁)⊕U(n₂)⊕···⊕U(nₖ). Euler's theorem, (a, n) = 1 then a^Φ(n)≡ 1 (mod n). Fermat, p prime, p ɫ a, ⇒ a^(p−1) ≡ 1 (mod p). ∀a, a^p ≡ a (mod p) Read more... 2290 words,🕔11 minutes read, May 14, 2022.

Cauchy's Theorem for Abelian Groups

G/Z(G)≋Inn(G). Cauchy's Theorem for Abelian Groups, G a finite Abelian group, |G| = n, p prime, p | n ⇒ G contains an element of order p. |G| = p², p prime ⇒ G ≈ ℤₚ² or ℤₚ ⊕ ℤₚ and G is Abelian. Read more... 1724 words,🕔9 minutes read, May 14, 2022.

Internal Direct Product

Internal Direct Product, G = H x K ↭ H and K ◁ G, H ∩ K = {e}, G = HK. H₁ x H₂ x ··· x Hₙ ≋ H₁ ⊕ H₂ ⊕ ··· ⊕ Hₙ. Read more... 1533 words,🕔8 minutes read, May 14, 2022.

Normal Subgroups

N ◁ G ↭ ∀a∈G, aH = Ha ↭ gHg⁻¹ ⊆ H ↭ gHg⁻¹ = H. Every subgroup of an Abelian group is normal. |G:H|=2 ⇒ H ◁ G. Z(G) ◁ G. Aₙ ◁ Sₙ. H ◁ G ⇒ G/H = {aH | a ∈ G} ≤ G, G/H is the factor or quotient group of G and H of order [G:H]. Read more... 2644 words,🕔13 minutes read, May 14, 2022.

Normal Subgroups II

N ◁ G ↭ ∀a∈G, aH = Ha ↭ gHg⁻¹ ⊆ H ↭ gHg⁻¹ = H. A₄ has no subgroups of order 6. If G/Z(G) is cyclic ⇒ G is Abelian. Let G be a non-Abelian group, |G| = pq, p and q primes ⇒ Z(G) = {e}. Read more... 2696 words,🕔13 minutes read, May 14, 2022.

Quaternion group

The quaternion group is a non Abelian group. Q₈ ≤ GL₂(ℂ). Q₈ = {1, -1, i, -i, j, -j, k, -k} = ⟨-1, i, j, k | (-1⟩² = 1, i² = j² = k² = ijk = -1⟩. Z(Q₈) = ⟨-1⟩ = {1, -1} ◁ Q₈. Every proper subgroup is Abelian, ⟨i⟩ ◁ Q₈. Q₈/⟨i⟩ = {⟨i⟩, j⟨i⟩} ≋ ℤ₂ Read more... 988 words,🕔5 minutes read, May 14, 2022.

Group Homomorphism

Φ is homomorphism ↭ Φ(a·b)=Φ(a)·Φ(b). Φ(e) = e'. Φ(aⁿ) = Φ(a)ⁿ. |Φ(a)| divides |a|. Ker(Φ) ≤ G. Φ(a) = Φ(b) ↭ aKer(Φ) = bKer(Φ). If Φ(a) = a' ⇒ Φ⁻¹(a') = aKer(Φ). Φ(H) ≤ G'. H is cyclic/Abelian ⇒ Φ(H) cyclic/Abelian. H ◁ G ⇒ Φ(H) ◁ Φ(G). Read more... 3753 words,🕔18 minutes read, May 14, 2022.

Group Homomorphism II

Φ is injective ↭ Ker(Φ) = {e}. If |Ker(Φ)| = n ⇒ Φ is an n-to-1 mapping from G onto Φ(G). |H| = n ⇒ |Φ(H)| | n. K ≤ G' ⇒ Φ⁻¹(K) ≤ G. K ◁ G' ⇒ Φ⁻¹(K) ◁ G, and in particular Ker(Φ) ◁ G. Φ onto and Ker(Φ) = {e} ⇒ Φ is an isomorphism. Read more... 1813 words,🕔9 minutes read, May 14, 2022.

Rotations of a regular tetrahedron

The group of rotational symmetries of a tetrahedron is isomorphic to the alternating group of even permutations on four elements, A4. Read more... 1038 words,🕔5 minutes read, May 14, 2022.

Abelian Groups. Fundamental Theorem of Finite Abelian Groups

Cauchy's Theorem. p prime, p| |G|, G Abelian ⇒ G has an element of order p. G finite Abelian p-group (∀g∈G, |g| = pᵏ for some k) ↭ |G| = pⁿ for some n. G finite Abelian group, m = |G| = p₁ʳ¹ ⊕ p₂ʳ² ⊕ ··· ⊕ pₖʳᵏ where p₁, p₂, ···, pᵣ distinct primes that divides m ⇒ G ≋ G₁ x G₂ x ··· Gᵣ with |Gᵢ| = pᵢʳⁱ Read more... 1761 words,🕔9 minutes read, May 14, 2022.

First Isomorphism Theorem

1st Isomorphism, Φ homomorphism, G/Ker(Φ) ≋ Φ(G), defined by gKer(Φ) → Φ(g). If Φ is a homomorphism from a finite group G to G' ⇒ |Φ(G)| divides |G| and |G'| Read more... 1893 words,🕔9 minutes read, May 14, 2022.

Fundamental Theorem of Finite Abelian Groups II

A finite, then A ⊕ B ≋ A ⊕ C iff B ≋ C. Every finite Abelian group is isomorphic to ℤₚ₁ⁿ¹ ⊕ ℤₚ₂ⁿ² ⊕ ··· ⊕ ℤₚₖⁿᵏ Read more... 1729 words,🕔9 minutes read, May 14, 2022.

Fundamental Theorem of Finite Abelian Groups. Exercises.

A finite, then A ⊕ B ≋ A ⊕ C iff B ≋ C. Every finite Abelian group is isomorphic to ℤₚ₁ⁿ¹ ⊕ ℤₚ₂ⁿ² ⊕ ··· ⊕ ℤₚₖⁿᵏ Read more... 1510 words,🕔8 minutes read, May 14, 2022.

The Normalizer/Centralizer Theorem.

The Normalizer/Centralizer Theorem. H ≤ G. Consider the mapping γ, N(H) → Aut(H), x → Φₓ, where Φₓ is the inner automorphism induced by x, Φₓ(h) = ghg⁻¹ ⇒ C(H) ◁ N(H) and N(H)/C(H) ≋ Aut(H). Every normal subgroup is the kernel of a homomorphism of G. Read more... 1433 words,🕔7 minutes read, May 14, 2022.

Rings

Ring (R, +, ·) = close (+, ·) + (R, +) Abelian group + Associative + Distributivity. A ring with unity (GL(n, ℝ)) = (R, ·), 1 ∈ R. Commutative ring (2ℤ) = (R, ·) commutative. Integral Domain (ℤ) = Commutative ring with unity + ab = 0 ⇒ a = 0 or b= 0. Division ring (Q₈) = Ring with unity, ∀a∈R, a≠0, ∃a⁻¹ s.t. a⁻¹a = aa⁻¹ = 1. Field = Commutative division ring. Read more... 1974 words,🕔10 minutes read, May 14, 2022.

Rings II. Properties. Subrings.

a0 = 0a = 0. a(-b) = (-a)b = -(ab). -(-a) = a. -(a +b) = -a + (-b). -(a -b) = -a + b. (-a)(-b) = ab. a(b - c) = ab - ac, (b-c)a = ba - ca. If R has unity, then (-1)a = -a, (-1)(-1) = 1, commutative, and it is unique. If an element has a multiplicative inverse, then it is unique. R is commutative ↭ (a+b)² = a² +2ab + b². Every boolean ring (∀a∈R, a² = a) is commutative. If a has a multiplicative inverse ⇒ a is not a zero divisor. Read more... 1951 words,🕔10 minutes read, May 14, 2022.

Ideals and Factor Rings

Ideal, ∀r ∈ R, a ∈ A, ra ∈ A and ar ∈ A ↭ a-b ∈ A and ra ∈ A, ar ∈ A. R commutative ring with unity, ⟨a⟩ = R ↭ a ∈ R is a unit. A principal ideal is an ideal generated by a single element ⟨a⟩ = {ra| r ∈ R}. Every ideal of ℤ is a principal ideal. R ring, I subring, R/I = {r + I | r ∈ R} is a ring ↭ I is an ideal of R. I, J ideals ⇒ I ∩ J, I + J, IJ are ideals. Read more... 3758 words,🕔18 minutes read, May 14, 2022.

Ideals. Exercises about rings

An element a ∈ R is nilpotent if xⁿ = 0 for some n ∈ ℕ, n > 0. R commutative ring, A ⊆ R, the annihilator of A is Ann(A) = {r ∈ R| ∀a ∈ A, r·a = 0}. R commutative ring, ideal I ⊆ R, the radical of I, rad(I) = √I = {r ∈ R | ∃n ∈ ℕ, rⁿ ∈ I}. If ⟨n⟩ is a principal ideal of the ring ℤ, n = p₁ᵏ¹ p₂ᵏ²··· pᵣᵏʳ, √⟨n⟩ = ⟨p₁·p₂···pᵣ⟩. Read more... 1958 words,🕔10 minutes read, May 14, 2022.

Prime and Maximal ideals

Prime ideal, ∀a, b ∈ R, ab ∈ A ⇒ a ∈ A or b ∈ A. Maximal ideal A, ∀B ideal, A ⊆ B ⊆ R ⇒ B = A or B = R. The prime ideals of ℤ are {0} and pℤ where p is prime. R commutative ring with unity, A ideal ⇒ A is prime ↭ R/A integral domain. R commutative ring with unity, A ideal ⇒ R/A is a field ↭ A is maximal. Read more... 3492 words,🕔17 minutes read, May 14, 2022.

Characteristic of a ring.

char(R) = the smallest n, nx = x + x + ··ₙ· + x = 0 ∀x ∈ R, char(ℤ) = char(ℚ) = char(ℝ) = char(ℝ) = 0, char(ℤₙ) = n. R ring with unity, char(R) = ord(1). Integral multiple of ring, ∀ x ∈ R, (m·x)○(n·x) = (m·n)·(x○x). D integral domain, char(D) = 0 or p. R finite, char(R) | |R|. Read more... 1908 words,🕔9 minutes read, May 14, 2022.

Integral domains.

An integral domain is a commutative ring with unity and no zero-divisors, ℤₚ, p prime. If R integral domain, a ≠ 0 and ab = ac ⇒ b = c. Every field is an integral domain. Every finite integral domain is a field. Read more... 1806 words,🕔9 minutes read, May 14, 2022.

Ring Homomorphisms

Φ ring homomorphism ↭ Φ(a + b) = Φ(a) + Φ(b), Φ(ab) = Φ(a)Φ(b), Φ(1) = 1. Φ(0) = 0, Φ(-1) = -1, Φ(nr) = nΦ(r), Φ(rⁿ) = (Φ(r))ⁿ, Φ(-r) = -Φ(r), ∀A subring of R, Φ(A) is subring of S. B ideal of S, Φ⁻¹(B) is ideal of R. R commutative ⇒ Φ(R) commutative. Φ onto, 1 ∈ R ⇒ Φ(1) is unit S. Φ injective ↭ Ker(Φ) = {0}. Φ isomorphism ⇒ Φ⁻¹ isomorphism. Ker(Φ) ideal in R. Read more... 3147 words,🕔15 minutes read, May 14, 2022.

Isomorphism Theorems for Rings

(R, +, ·) ring, A ideal of R. (R/A, +, ·) is the quotient ring of R by A. Every ideal is the kernel of a ring homomorphism. 1st Theorem, Φ ring homomorphism, R/Ker(Φ) ≋ Φ(R). 2nd. I subring and J ideal of R ⇒ I+J subring, I∩J ideal of I, I/I∩J ≋ I+J/J. Read more... 1770 words,🕔9 minutes read, May 14, 2022.

Isomorphism Theorems for Rings II

There is a unique ring homomorphism from the ring of integers to any ring. Every ring has a subring isomorphic to either ℤ or ℤn. Every ring has a subring isomorphic to either ℤp or ℚ. 3rd. isomorphism theorem, R a ring with ideals I ⊆ J ⊆ R, (R/I)/(J/I) ≋ R/J Read more... 2006 words,🕔10 minutes read, May 14, 2022.

Polynomial Rings. Division Algorithm.

R[x] is a ring. D commutative ring with 1, D integral domain ↭ D[x] integral domain. R integral domain, f, g nonzero polynomials in R[x] ⇒ deg(fg) = deg(f) + deg(g). R integral domain ⇒ units of R[x] = units of R. I ideal of R, R[x]/I[x] ≋ (R/I)[x]. P ⊆ R, P prime ideal ↭ P[x] ⊆ R[x], P[x] prime ideal. Division Algorithm. F field, ∀f(x), g(x) ∈ F[x], g(x) ≠ 0 ⇒ ∃!q(x), r(x) ∈ F[x] s.t. f(x) = g(x)q(x) + r(x) and either r(x) = 0 or deg(r(x)) < deg(g(x)). Read more... 2238 words,🕔11 minutes read, May 14, 2022.

The Field of Quotients and More Exercises

Let D be an integral domain. Then, there exists a field F that contains a subring isomorphic to D. ℚ[√2] is a field. Find ring isomorphisms Φ, ℚ[√3] → ℚ[√3]. The quadratic fields ℚ[√2] and ℚ[√3] are not isomorphic. Read more... 2659 words,🕔13 minutes read, May 14, 2022.

Polynomial Rings II. Principal ideal domains.

g | f ↭ r = 0. Factor Theorem, F field, f ∈ F[x], α ∈ F zero or root of f (f(α) = 0) ↭ (x -α)|f(x) ↭ ∃q ∈ F[x], f(x) = (x - α)q(x). def(f) = n ⇒ f has at most n distinct zeros in F, counting multiplicity. Principal ideal ⟨a⟩ = Ra = {ra | r ∈ R}. R is PID ↭ every ideal is principal, e.g., ℤ, F (⟨0⟩, ⟨1⟩ = F) and F[x] (F field) Read more... 2576 words,🕔13 minutes read, May 14, 2022.

⟨p(x)⟩ maximal iff irreducible. Unique Factorization in ℤ[x]

F field, p(x) ∈ F[x]. ⟨p(x)⟩ is a maximal ideal in F[x] ↭ p(x) is irreducible over F. p(x) irreducible over F ↭ F[x]/⟨p(x)⟩ field. If p(x) irreducible over F field, p(x)|a(x)b(x) ⇒ p(x)|a(x) or p(x)|b(x). p(x) ∈ ℤ[x], p(x) =[unique factorization] b₁b₂···bₛp₁(x)p₂(x)···pₘ(x), where bᵢ's irreducible polynomials of degree 0, and the pᵢ(x)'s irreducible polynomials of positive degree. Read more... 2329 words,🕔11 minutes read, May 14, 2022.

Factorization of Polynomials

F field, f(x)∈F[x], deg(f) = 2/3 ⇒ f reducible ↭ f has a zero in F. content(aₙxⁿ+aₙ₋₁xⁿ⁻¹+···a₀) = gcd(aₙ, aₙ₋₁,···, a₀). Gauss Lemma. D UFD, f(x), g(x) ∈ D[x], content(f·g) = content(f)·content(g). Corollary. D UFD, F = Frac(D), f ∈ D[x], ∃α, β ∈ F[x], f = α·β ⇒ ∃a, b ∈ D[x], deg(α) = deg(a), deg(β) = deg(b), and f(x) = a·b. A non-constant p ∈ ℤ[x] is irreducible in ℤ[x] ↭ p is irreducible in ℚ[x] and primitive in ℤ[x]. D is a UFD ↭ D[x] is a UFD Read more... 2222 words,🕔11 minutes read, May 14, 2022.

Irreducibility Tests

Mod p Test, f ∈ ℤ[x], f' ∈ ℤₚ[x] by reducing coefficients mod p. f' irreducible over ℤₚ & deg(f') = deg(f) ⇒ f irreducible over ℚ. Eisenstein Criterion, p prime, f(x) = aₙxⁿ+aₙ₋₁xⁿ⁻¹+···+ a₀ ∈ ℤ[x], deg(f) ≥ 1, p|a₀, p|a₁,···, p|aₙ−₁, pɫaₙ, p²ɫa₀ ⇒ f irreducible ℚ. Φₚ(x) =xᵖ⁻¹/x-1 = xᵖ⁻¹+xᵖ⁻²+··· +1 irreducible ℚ. Rational root test, r/s rational solution of f satisfies r|a₀ and s|aₙ Read more... 2141 words,🕔11 minutes read, May 14, 2022.

Divisibility in Integral Domains.

a|b if ∃c∈R, b = ac. a & b are associates if a = ub, u = unit of D. a is irreducible if when a = bc ⇒ b or c is a unit. a is prime if when a|bc ⇒ a|b or a|c. D integral domain, prime ⇒ irreducible. PID, irreducible ↭ prime. ℤ, F[x], where F field are PID, but ℤ[x] is not PID. Read more... 2248 words,🕔11 minutes read, May 14, 2022.

Euclidean and Noetherian domain

Noetherian domain = integral domain + ascending chain condition, I₁⊆ I₂ ⊆ ··· ∃N ∈ ℕ, Iₙ = Iɴ, ∀n ≥ ɴ. Every PID satisfies the ascending chain condition. Every PID is a UFD. Let F be a field ⇒ F[x] is a UFD. Euclidean domain is an integral domain + (i)∀a,b ∈ D, a≠0, b≠0, N(a) ≤ N(ab), (ii)∀a,b ∈ D, b≠0, ∃q,r ∈D, a = bq + r, r = 0 or N(r) < N(b). Euclidean Domain ⇒ PID ⇒ UFD. Read more... 2883 words,🕔14 minutes read, May 14, 2022.

Unique Factorization Domains

An integral domain D is a UFD if every non-zero non-unit element of D can be written uniquely (order and units) as a product of irreducibles. D PID, (i) a|b ↭ ⟨b⟩ ⊆ ⟨a⟩; (ii) a and b are associates ↭ ⟨b⟩ = ⟨a⟩; (iii) a is a unit ↭ ⟨a⟩ = D. R UFD, a ∈ R, a irreducible ↭ a prime. D PID, ⟨p⟩ is a maximal ideal ↭ p is irreducible. D PID, p ∈ D irreducible ⇒ prime. Read more... 1743 words,🕔9 minutes read, May 14, 2022.

Vector Spaces

V is a vector space over F if (V, +) is an Abelian group and ∀a, b ∈ F, u, v ∈V, a(v + u) = av + au, (a + b)v = av + bv; a(bv) = (ab)v; 1v = v. U is a subspace of V is U is also a vector space over F under the operations of V ↭ (U, +)≤(V, +), ∀u∈U,α∈F, αu∈U ↭ αu, u + v ∈ U. Subspace of V spanned by v₁, v₂, ···, vₙ, span{v₁, v₂, ···, vₙ} = {α₁v₁ + α₂v₂ + ··· + αₙvₙ| αᵢ ∈ F}. Read more... 1522 words,🕔8 minutes read, May 14, 2022.

Vector Spaces II

S is linearly independent if α₁v₁ + α₂v₂ + ··· + αₙvₙ = 0 ⇒ α₁ = α₂ = ··· = αₙ = 0. S l.i. ↭ ∃vᵢ linear combination of the others. B basis if linearly independent and span{v₁, v₂, ···, vₙ} = V. B is a basis of V ↭ ∀v ∈ V can be written uniquely as a linear combination of elements of the basis. If B = {v₁, v₂, ···, vₙ} is a basis for V, dim(V) =n. Every basis has the same number of vectors. Read more... 2063 words,🕔10 minutes read, May 14, 2022.

Field Theory

char(F) = the smallest p s.t. p·1 = 1+1+··ₚ··+1= 0 if such a p exists, or char(F) = 0. char(F) = 0 or p, prime. F ⊆ E ⊆ K tower of fields. If [K:E] < ∞ & [E:F] < ∞ ⇒ [K:F] < ∞ & [K:F] =[K:E][E:F]. σ :F → E embedding of fields ⇒ F ⊆ K, σ can be extended to an isomorphism σ* :K → E. Krnoecker's Theorem. F field, f(x) non-constant polynomial in F[x] ⇒ ∃ an extension E of F in which f(x) has a root. Th, p(x) ∈ F[x] irreducible, u root of p(x) in an extension E of F ⇒ F(u) = F[u] = {b₀ + b₁u + ··· + bₘuᵐ ∈ E | b₀ + b₁x + ··· + bₘxᵐ ∈ F[x]}. If degree(p) = n, (1, u, ···, uⁿ⁻¹) forms a basis of F(u) over F and [F(u):F] = n. Read more... 3336 words,🕔16 minutes read, May 14, 2022.

Field Theory II

α ∈ E is algebraic over F if ∃p(x) ∈ F[x] s.t. p(α) = 0. E/F is algebraic if ∀e ∈ E, e is algebraic over F. E/F extension field, u ∈ E algebraic over F, p(x) ∈ F[x] be a polynomial of least degree such that p(u) = 0 (the minimal polynomial of u over E) ⇒ p(x) is irreducible over F; if g(x) ∈ F[x], g(u) = 0 ⇒ p(x) | g(x); ∃! monic polynomial p(x) ∈ F[x] of least degree such that p(u) = 0. If E/F finite extension ⇒ E/F algebraic extension. Th. E/F extension, α∈ E algebraic over F ⇒ F(α)/F algebraic extension, [F(α):F)] = n = deg(p(x)) where p is the minimal polynomial of α. Read more... 1947 words,🕔10 minutes read, May 14, 2022.

Splitting Fields

K/F, L/F extension fields, α ∈ K algebraic over F, σ :K → L F-homomorphism of fields ⇒ σ(α) is algebraic over F. f(x) ∈ F[x], K/F s.f. if f(x) = c(x -α₁)(x -α₂) ··· (x -αₙ), αᵢ ∈ K, K = F(α₁, ···, αₙ). Existence of Splitting Fields, F field, f(x) ∈ F[x] ⇒ ∃ s.f. E for f(x) over F. Th, F field, p(x) ∈ F[x] irreducible polynomial over F, deg(p(x)) = n. If a is a root of p(x) in some extension E of F, then F(a) ≋ F[x]/⟨p(x)⟩ & F(a) is a vector space over F with a basis {1, a, ···, aⁿ⁻¹}. Degree of Splitting Field. F field, p ∈ F[x], deg(p) = n ⇒ ∃K/F s.f., [K:F] ≤ n! Read more... 3134 words,🕔15 minutes read, May 14, 2022.

Extension Theorems

ϕ:E→F isom, K/E, α ∈ K algebraic over E with minimal polynomial p(x). L/F, β ∈ L root of Φ(p(x)) in F[x] ⇒ ϕ extends to !isomorphism ϕ':E(α)→F(β), ϕ'(α)=β, ϕ' agrees with Φ on E. Th. ϕ:E→F isom, p(x)∈E[x], p(x)→q(x)∈F[x]. K, L s.f. of p(x) and q(x) ⇒ ϕ extends to an isomorphism ψ:K→L, ψ agrees with ϕ on E. Th. p(x)∈F[x] ⇒ ∃! (up to isomorphism) K s.f. of p(x). Th. E = F(α)/F, α algebraic over F, f ∈ F[x] irred. α over F, Φ:F → K homom. If β ∈ L (L/K) root of Φ(f) ⇒ ∃! extension of Φ, Φ':E→L such that ϕ'(α) = β and fixes F. Read more... 1709 words,🕔9 minutes read, May 14, 2022.

Extension Theorems II

L/K algebraic extension, every Φ:K→C field homomorphism where C is an algebraically closed field, can be extended to a homomorphism L → C. Th. i:K→L homomorphism s.t. L/i(K) is an algebraic extension. If ϕ:K→C field homomorphism & C is an algebraically closed field ⇒ ∃ σ:L→C field homomorphism s.t. σ ◦ i = ϕ. F is algebraically closed ↭ ∀f(x) ∈ F[x], f factors into linear polynomials. An algebraically closed field has no proper algebraic extensions. All algebraic closures of a field are isomorphic. Th. Every F-homomorphism between K to itself is an isomorphism. Read more... 2207 words,🕔11 minutes read, May 14, 2022.

Algebraic Extensions. Characterization of field extensions.

α ∈ E algebraic, E/F ⇒ its minimal polynomial is prime. Characterization of field extensions, E/F, α ∈ E. α transcendental over F then F(α) ≋ F(x). α algebraic over F ⇒ F(α) ≋ F[x]/⟨p(x)⟩ where p(x) is a unique monic irreducible over F (ker(Φ)=⟨p(x⟩), and a polynomial in F[x] of minimum degree such that p(α) = 0. Th, L/K, α ∈ L, α is algebraic over K ↭ α is in a finite extension of K. Th. F field, p(x) ∈ F[x] - F ⇒ the ring F[x]/p(x) is a field ↭ p is irreducible Read more... 1837 words,🕔9 minutes read, May 14, 2022.

Algebraic Extensions II. Primitive Element Theorem

Primitive Element Theorem, F field, char(F) = 0, a and b algebraic over F ⇒ ∃c ∈ F(a, b) s.t. F(a, b) = F(c). Algebraic over Algebraic is Algebraic. If K is an algebraic extension of E and E is an algebraic extension of F ⇒ K is an algebraic extension of F. Given a field extension E/F ⇒ the elements of E that are algebraic over F form a subfield (the algebraic closure of F in E). A field K is algebraically closed if ∀f(x) ∈ K[x] \ K has a root in K ↭ ∀f(x) ∈ K[x] irreducible, f has degree 1. Th, f(x)∈F[x], deg(f)=n. K s.f. f(x) over F ⇒ [K:F] ≤ n! Read more... 1795 words,🕔9 minutes read, Apr 14, 2022.

Finite Fields

A field is a prime field ↭ it has no proper subfields. The characteristic of a finite field has to be a prime number. Th, F finite field of characteristic p ⇒ |F| = pⁿ. Every field F is either an extension of ℚ or ℤₚ, p prime. Freshman's Dream or Exponentiation, R commutative ring with unity of characteristic p ⇒ (a + b)ᵖ^ⁿ = aᵖ^ⁿ + bᵖ^ⁿ. f(x) ∈ F[x] is separable if its roots are distinct in the s.f. of f(x). E/F separable, ∀a ∈ E, a is the root of a separable polynomial in F[x]. Th, F field, f(x) ∈ F[x] ⇒ f is separable ↭ (f, f') = 1. Read more... 1680 words,🕔8 minutes read, May 14, 2022.

Finite Fields 2

Classification of finite fields. There exist a unique finite field F of order pⁿ, up to isomorphism & F ≋ s.f. xᵖ^ⁿ-x over ℤₚ. A field of order pⁿ contains a field of order pᵐ iff m|n. Th. F field, G a finite subgroup of F* ⇒ G is cyclic. In particular, the multiplicative group of all nonzero elements of a finite field F* is cyclic. Read more... 1860 words,🕔9 minutes read, May 14, 2022.

Conjugacy Classes. The Class Equation.

a,b ∈ G conjugate if xax⁻¹ = b. cl(a) = {xax⁻¹ | x ∈ G}. If G is finite ⇒ |cl(g)| = |G:Z(g)|. If G is finite group ⇒ |cl(a)| | |G|. The Class Equation. G a nontrivial finite group. a₁, a₂, ···, aₖ are the representatives of the conjugacy classes that have size > 1 ⇒ |G| = |Z(G)|+ Σₖ|G:C(aₖ)|. Every p-group has a nontrivial center. If G/Z(G) is cyclic ⇒ G is Abelian. If |G| = p², p prime ⇒ G is Abelian. Read more... 1040 words,🕔5 minutes read, May 14, 2022.

Impossible Geometric Constructions.

α is constructible if and only if there is a tower of fields ℚ = F₀ ⊆ F₁ ⊆···⊆ Fₖ ⊆ℝ such that α ∈ Fₖ & and Fᵢ = Fᵢ−₁(√aᵢ). If α is constructable ⇒ [ℚ(α):ℚ] = 2ᵏ. It is not possible to square the circle, double the cube, and trisect an angle. Read more... 1697 words,🕔8 minutes read, May 14, 2022.

Cauchy and First Sylow Theorem.

Sylow’s First Theorem. G finite group, p prime. If pᵏ | |G| for k ∈ ℤ ⇒ ∃H ≤ G, |H| = pᵏ. A Sylow p-subgroup of G is a maximal p-subgroup of G, that is, pᵏ divides |G| and pᵏ⁺¹ does not divide |G|. Cauchy's Theorem. Let G be a finite group, p prime, p | |G| ⇒ ∃a ∈ G such that |a| = p. Read more... 1098 words,🕔6 minutes read, Jan 01, 0001.

Group Actions. The Fundamental Counting Principal.

X set, G group. A left group action of G on X is a map G x X → X, (g, x) → g·x s.t. (i) Identity. ∀x ∈ X, e·x = x; (ii) Compatibility. ∀x ∈ X, ∀g, h ∈ G, g·(h·x) = (gh)·x. A group G acts on itself by conjugation (g·x = gxg⁻¹). Let X be a G-set. ∀x, y ∈ X, x ~ y ↭ ∃g ∈ G s.t. g·x = y ⇒ ~ is an equivalence relation on X (G-equivalence). The equivalence classes are called orbits, where Oₓ is the orbit containing x. The Fundamental Counting Principle. G be a group acting on X, x ∈ X ⇒ |Oₓ| = [G:Gₓ] Read more... 1193 words,🕔6 minutes read, May 14, 2022.

Some applications of Sylow’s Theorems

G has an unique subgroup H of a given order ⇒ H ◁ G. Corollary. A Sylow p-subgroup of a finite group G is normal ↭ it is the only Sylow p-subgroup. Th. H, K ◁ G, H ∩ K = {e}, and HK = G ⇒ G ≋ H x K. Th. p odd prime ⇒ there are only two groups of order 2p, ℤ₂ₚ and Dₚ. Read more... 2081 words,🕔10 minutes read, May 14, 2022.

Sylow Theorem II and III.

P Sylow p-subgroup, x ∈ G an element whose order is a power of p. If x ∈ N(P) ⇒ x ∈ P. Th, H, K ≤ G. The number of distinct H-conjugates of K = [H:N(K) ∩ H]. Sylow's Second Theorem. G finite, p | |G| ⇒ all Sylow p-subgroups are conjugates. Sylow's Third Theorem. G finite, p | |G| ⇒ the number of Sylow p-subgroups of G is congruent to 1 (mod p) and divides |G|. Read more... 2254 words,🕔11 minutes read, May 14, 2022.

Simple Groups

G simple ↭ it has no trivial, proper normal subgroups. H ≤ G Abelian ⇒ H ◁ G. G Abelian ⇒ G simple ↭ G is cyclic of primer order. N ◁ G & H ≤ G ⇒ N ∩ H ◁ H. ∀n ≥ 3, Aₙ is generated by the set of 3-cycles. ∀n ≥ 5, all 3-cycles are conjugates. Aₙ, n ≥ 5 is a simple group. Read more... 1801 words,🕔9 minutes read, May 14, 2022.

The Hunt for Non-Abelian Simple Groups.

A group G is simple if it has no trivial, proper normal subgroups. |G| = p, prime ⇒ G is simple. A finite p-group (|G| = pⁿ) cannot be simple unless it has order p. |G| = p·q, p and q are distinct primes ⇒ G is not simple. Read more... 2323 words,🕔11 minutes read, May 14, 2022.

Symmetric Key Cryptography

Cryptography is the study of techniques to secretly obscure, store or communicate messages so outsiders cannot read the message. Symmetric key cryptography is any cryptographic algorithm that is based on a shared key that is used to encrypt or decrypt text. Caesar cipher is one of the earliest and simplest methods of encryption technique where each letter is replaced by another letter some fixed number of positions later in the alphabet. Public Key Cryptography. Read more... 2005 words,🕔10 minutes read, May 14, 2022.

Generators and Relations

The set of all equivalence classes of elements of W(S) is called the free group. The universal mapping property. Every group is a homomorphic image of a free group. Dick's theorem. G = ⟨a₁, a₂,..., aₙ | w₁ = w₂ = ... = wₜ = e⟩ and let create a new group by imposing additional relations, G' = ⟨a₁, a₂,..., aₙ | w₁ = w₂ = ... = wₜ = wₜ+₁ = ··· = wₜ+ₖ = e⟩ ⇒ G ≋ G' Read more... 3086 words,🕔15 minutes read, May 14, 2022.

Symmetry Groups

Symmetry Groups. Classification of isometries. Any planar isometry is either a rotation, a translation, a reflection about a line or a glide reflection. Every finite group G of isometries of the plane is isomorphic to ℤₙ or Dₙ. Read more... 3153 words,🕔15 minutes read, May 14, 2022.

Frieze and Wallpaper Groups

Frieze Groups. Wallpaper or crystallographic group. The collection of discrete frieze groups consists precisely of the following four groups, ℤ, ℤ⊕ℤ₂, D∞, and D∞⊕ℤ₂. There are 17 wallpaper symmetry groups. Read more... 1287 words,🕔7 minutes read, May 14, 2022.

Coding Theory

Binary Symmetric Channel. ASCII. The Hamming Metric. The Hamming distance is a metric in Fⁿ. Systematic code. Parity-Check Matrix Decoding. Orthogonality Lemma. Let C be a systematic (n, k) linear code over F ⇒ ∀v ∈ Fⁿ, v ∈ C ↭ vH = 0. Parity-check matrix decoding will correct any single error iff if the rows of the parity-check matrix are nonzero and no one row is a scalar multiple of any other. Coset Decoding. The coset decoding is the same as nearest-neighbor decoding. Read more... 5456 words,🕔26 minutes read, May 14, 2022.

Symmetry and Counting

The set of orbits of X under the action of G forms a partition of X. The stabilizers of elements in the same orbit are conjugate to each other. Burnside's Theorem. Read more... 2077 words,🕔10 minutes read, May 14, 2022.

Cayley Diagraphs. Hamiltonian paths.

Cay({(1, 0), (0, 1) | ℤₘ⊕ℤₙ) does not have a Hamiltonian circuit when m, n > 1 and gcd(m, n) = 1. Cay({(1, 0), (0, 1) | ℤₘ⊕ℤₙ) has a Hamiltonian circuit when n divides m. Let G be a finite Abelian group and S any non-empty generating set for G. Then, the Cayley diagraph of G, Cay(S:G), has a Hamiltonian path. Read more... 1268 words,🕔6 minutes read, May 14, 2022.

Group Characters. Fixed Fields

Fixed fields. Character of a group. Independence of field homomorphisms. Mutually distinct characters of a group in a field are independent. Frobenius homomorphism. Read more... 2220 words,🕔11 minutes read, May 14, 2022.

Theorem on fixed fields

Degree of the fixed field extension is at least the number of homomorphisms Read more... 1914 words,🕔9 minutes read, May 14, 2022.

Automorphisms. Galois extensions and groups

The Galois group of E over F is the set of all automorphisms of E which fix F. E/F, f(x) a minimal polynomial of α ∈ E ⇒ Any automorphism σ ∈ Gal(E/F) induces a permutation of the set of zeros of f. Order of Galois Group equals degree of extension. Read more... 2361 words,🕔12 minutes read, May 14, 2022.

Galois extensions and groups II

Galois extensions. Galois groups. If F is the fixed field of G ⇒ G = Gal(K/F) = Gal(K/K^G). Read more... 1432 words,🕔7 minutes read, May 14, 2022.

Normal extensions

Equivalence of Definition of Normal Extensions. A field extension of degree 2 is a normal extension. Read more... 2281 words,🕔11 minutes read, May 14, 2022.

Normal extensions 2

K/F normal, F ⊂ L ⊂ K ⇒ K/L normal and L/F is normal ↭ if σ(L) ⊆ L. Intersection and composite are normal. E/F finite extension, then exists a normal closure N. Read more... 2611 words,🕔13 minutes read, May 14, 2022.

Computing Galois group. Determine Galois and normal extension.

Exercises. ℚ(i), x^4+4, ℚ(√2, i), Q(ζ8), ℚ(√3,√5), ℚ(∛2), ℚ(∛5, √7), ℚ(∜5). Read more... 2126 words,🕔10 minutes read, May 14, 2022.

Separable extensions

Derivate of a polynomial. Zeros of an irreducible polynomial. Criterion for multiples zeros. Bezout's identity in F[x]. Perfect fields. f is separable ↭ (f, f') = 1. If char(F) = 0, F is a finite field, or perfect ⇒ every polynomial in F[x] is separable. Read more... 2299 words,🕔11 minutes read, May 14, 2022.

Computing Galois group II. Determine Galois and normal extension.

Exercises. ℚ(√2,√3), ℚ(∛2, w), ℚ(√2,√3,√5), ℚ(∜2), Fq. Read more... 2781 words,🕔14 minutes read, May 14, 2022.

Separable extensions II

In positive characteristic fields, there exist inseparable irreducible polynomial. Separability of towers (F ⊆ K ⊆ L), L/F is separable ↭ L/K and K/F are separable. Read more... 2283 words,🕔11 minutes read, May 14, 2022.

Characterization of Galois extensions

K/F is Galois iff K is the splitting field of a separable polynomial over F Read more... 3366 words,🕔16 minutes read, May 14, 2022.

Galois extension II. Exercises.

F ⊆ L ⊆ K are field extensions, K/F Galois ⇒ K/L is Galois. Every finite separable field extension K/F can be extended to a Galois extension. A normal extension does not need to be Galois. If a polynomial has degree 2 or 3 and has no roots over a field F, then f is irreducible in F[x]. Read more... 1302 words,🕔7 minutes read, May 14, 2022.

Motivating the fundamental theorem of Galois Theory

There is a bijection correspondence between the intermediate fields of the extension K/F and the Galois subgroups. All extensions of finite fields are cyclic. A normal extension does not need to be Galois. Read more... 2017 words,🕔10 minutes read, May 14, 2022.

Fundamental Theorem of Galois Theory

Main theorem of Galois theory. Read more... 1815 words,🕔9 minutes read, May 14, 2022.

Fundamental Theorem of Galois Theory II.

There are only a finitely many intermediate fields of finite field separable extensions. Read more... 1056 words,🕔5 minutes read, May 14, 2022.

Consequences Galois III. Exercises. ℚ(√2+√3+√5).

Let K/ℚ be a normal extension, |Gal(K/ℚ)|=8 and σ^2 = id ∀σ ∈Gal(K/ℚ), σ ≠ id. Find the number of intermediate fields. Read more... 2258 words,🕔11 minutes read, May 14, 2022.

Consequences Galois: Fundamental Theorem of Algebra

A degree 2 extension can be obtained by adding a square root of an element of F. Let K/F be a Galois extension such that G = Gal(K/F) ≋ S3 ⇒ K is s.f. of an irreducible cubic polynomial over F. L1, L2 Galois extensions ⇒ L1 ∩ L2 is also Galois. Read more... 1838 words,🕔9 minutes read, May 14, 2022.

Consequences Galois II. Exercises.

Splitting fields of x^3 -2 over F2, F3, F5 and F7. Read more... 1222 words,🕔6 minutes read, May 14, 2022.

Kummer extensions

K/F is a Kummer extension, i.e., ∃a ∈ F, x^n-a is an irreducible polynomial and K is the splitting ↭ K/F is a cyclic extension. Read more... 2416 words,🕔12 minutes read, May 14, 2022.

Cyclotomic extensions and polynomials

K/F s.f. x^n-1 over F, there exists an injective homomorphism Gal(K/F) → (ℤ/nℤ)*. K/ℚ is the s.f. of x^n -1, then Gal(K/F) ≋ (ℤ/nℤ)*. Abel's Theorem. Read more... 3709 words,🕔18 minutes read, May 14, 2022.

Solvability by radicals

(Simple) Radical extensions. A degree six Galois extension is solvable. Cyclic extensions are solvable. Read more... 1853 words,🕔9 minutes read, May 14, 2022.

Characterizations of solvability. Cubic polynomials.

Different characterizations of solvability, Abelian, cyclic extensions, roots of unity. Cubic polynomials. Read more... 2387 words,🕔12 minutes read, May 14, 2022.

Characterizations of solvability II. Discriminants.

Galois group of a polynomial. The Galois group of f is a subgroup of Sn. f is irreducible ↭ Gal(f) is transitive subgroup Sn. If δ ∈ F (Δ is a square in F), then G ⊆ An. Otherwise, G ⊊ An. Read more... 1832 words,🕔9 minutes read, May 14, 2022.

Quartics are solvable

Quartic polynomials are solvable. Discriminants. Resolvent cubics. g(x) ∈ F[x]. Disc(f) = Disc(g) and βi are all distinct. Read more... 2186 words,🕔11 minutes read, May 14, 2022.

Solvable groups. Galois' theorem.

Solving polynomials by radicals. Sn is not solvable n≥5. Galois' theorem (G = Gal(f) is solvable ↭ f is solvable -by radicals-). f polynomial, deg(f)≤4 ⇒ f is solvable. Read more... 2748 words,🕔13 minutes read, May 14, 2022.

Composite fields

The composite of two radical extensions is radical. The composite of Galois extensions is Galois. Degrees of Composites. A radical extension can be extended to be both Galois and radical. Read more... 1495 words,🕔8 minutes read, May 14, 2022.

Solvable groups II.

Burnside's Theorem for p-Groups. Every p-group is solvable. |G| = p^k·q^s, p and q prime numbers, and 1 mod p ≠ q^t for t = 1···s, then G is solvable Read more... 862 words,🕔5 minutes read, May 14, 2022.

Insolvability of quintics.

Let f be an irreducible polynomial of degree 5 over a subfield F of the complex numbers whose Galois group is either A5 or S5, then f is not solvable. Read more... 1683 words,🕔8 minutes read, May 14, 2022.

Insolvability of quintics II.

Sn can be generated by a n-cycle and a transposition. An irreducible polynomial of degree p over ℚ, p ≥ 5, with three real roots. Then, Gal(f) ≋ Sp, so it is not solvable. General quintic polynomials are not solvable by radicals. Read more... 1443 words,🕔7 minutes read, May 14, 2022.

Computing Galois groups. Degree 2 and 3.

If f irreducible quadratic with exactly two real roots, then Gal(f) = S4 or D4. If one root of f irreducible is solvable over F (⊆ ℂ), then f is solvable. Read more... 900 words,🕔5 minutes read, May 14, 2022.

Computing Galois groups II. Quartics and beyond.

If g splits completely in F ↭ G = D2. If g has exactly one root in F ↭ G = D4 or C4. g is irreducible ↭ G = S4 or A4. Resolvent cubic. Read more... 2039 words,🕔10 minutes read, May 14, 2022.

Every finite Abelian group is realized as a Galois group over ℚ.

Every finite Abelian group is realized as a Galois group over ℚ. Find a degree 4 (n) extension of ℚ with no intermediate field. Every finite group is isomorphic to some Galois group for some finite normal extension of some field. Read more... 2167 words,🕔11 minutes read, May 14, 2022.

A5 is simple. Primitive element theorem.

All cycles of the same length in Sn are conjugate. A3 is generated by 3 cycles. All 3-cycles are conjugate in An. Primitive element theorem, K/F finite extension is primitive ↭ ∃ only finitely many subfields. A finite separable extension K/F is primitive. Read more... 1713 words,🕔9 minutes read, May 14, 2022.