A group is a nonempty set G together with a binary operation on G, denoted as ◦, ·, * or simply omitted, satisfying the following axioms:
The group is called Abelian if ab = ba ∀a, b ∈ G.
A group is finite if it has a finite number of elements. Otherwise, it is said to be infinite. The number of elements in a group G is called the order of the group, and is denoted by |G|, e.g., ℤ_{10} and U_{10} = {1, 3, 7, 9} -positive integers less than 10 that are coprime to 10- are finite group of orders 10 and 4 respectively. The order of an element a ∈ G is the smallest positive integer n such that x^{n} = 1, e.g., |0| = 1, |1| = 6, |2| = 3, |3| = 2, |4| = 3, |5| = 6 in Z_{6}.
Proposition. Let (G, ∘) be a group.
The group ℤ_{n} consists of the elements {0, 1,2, . . . , n−1} with addition mod n as the operation. You can also multiply elements of Z_{n}, but you do not obtain a group. For each n > 1, we define U_{n} to be the set of all positive integers less than n that have a multiplicative inverse modulo n, that is, there is an element b∈ Z_{n} such that ab = 1. U_{n} = {a ∈ Z_{n}: a is relatively prime to n, a < n} = {a ∈ Z_{n}: gcd(k, n)=1}.
The dihedral group D_{n} is the group of symmetries of a regular polygon with n vertices, which includes rotations and reflections or, alternatively, the right motions taking a regular n-gon back to itself. D_{n} = {e, r, r^{2},…, r^{n-1}, s, rs, r^{2}s,…, r^{n-1}s} = ⟨r, s | r^{n} = s^{2} = e, rs = sr^{n-1}⟩.
Definition. Let G be a group. A subgroup is a subset of a group that itself is a group. Formally, H ⊂ G (H is a subset of G), ∀a, b ∈ H, a·b ∈ H, and (H, ·) is a group (closure, identity, and inverses).
H ≤ G if H is closed (∀a, b ∈ H, a·b ∈ H) and every element in H has an inverse in H (∀a ∈ H, a^{-1} ∈ H). Besides, H ≤ G iff ∀a, b ∈ H, a·b^{-1} ∈ H.A non empty finite subset H ⊂ G closed under the operation of G (∀a, b ∈ H, a·b ∈ H) is a subgroup of G.
The center of a group, G, is the subset of elements that commute with every element of G. Z(G) = {a ∈ G | ab = ba ∀ b in G} ≤ G. Let G be a group, H ≤ G be a subgroup of G. Then the conjugate of H by g is a subgroup of G, that is, g^{-1}Hg = {g^{-1}hg | h ∈ H} ≤ G.
Let G be a group and a be any element in G (a ∈ G). Then, the set ⟨a⟩={a^{n} | n ∈ Z} is a subgroup of G. We call it the cyclic subgroup generated by a.
Definition. Let G be a group. A cyclic group G is a group that can be generated by a single element a. More formally, G is cyclic if there exists at least an element a ∈ G: G = ⟨a⟩ = {a^{n} | n ∈ ℤ}
Definition. Let G be a group, H ≤ G. A left coset of H in G is a subset of the form aH = [ah | h ∈ H] for some a ∈ G. This is a equivalent relation, so aH = [a], and a is the representative of the coset aH. The collection of left cosets is denoted by G/H.
Lemma. Let G be a group, H ≤ G, g_{1}, g_{2} ∈ G. The following statements are equivalent:
Definition. Let G be a group, H < G a subgroup. The set of cosets of H in G is denoted as G/H and called quotient set of G by H. G/H = {aH | a ∈ G}. The index of H in G is denoted as [G:H] = |G/H| where we say [G:H] = ∞ if G/H is infinite.
Let (A, *) and (B, ⋄) be two binary algebraic structures. A homomorphism is a structure-preserving map between two algebraic structures of the same type or, in other words, a map ϕ: A → B, such that ∀x, y ∈ S : ϕ(x ∗ y) = ϕ(x) ⋄ ϕ(y).
Let (A, *) and (B’, ⋄) be two binary algebraic structures. An isomorphism is a bijective homomorphism, i.e., one-to-one and onto.
Cayley’s Theorem. Every group G is isomorphic to a a subgroup of a symmetric group. In particular, every finite group is isomorphic to a subgroup of S_{n}. Lagrange’s Theorem. Let G be a finite group and let H ≤ G. Then, |H| | |G|. In particular, the order of an element (the order of the subgroup generated by a) of a finite group divides the order of the group.
Cauchy’s Theorem. Let G be a finite group and let p be a prime that divides the order of G. Then, G has an element of order p.
A group is called Abelian if ab = ba ∀a, b ∈ G. Every group of prime order (in fact, it is cyclic), order five or smaller, or cyclic is Abelian. $\mathbb{S}_3$ is the smallest non-commutative group.
A subgroup H of a group G is called a normal subgroup of G if aH = Ha, ∀a ∈ G. The alternating group of even permutations A_{3} = {(), (123) -(13)(12)-, (132) -(12)(13)-} is a normal subgroup of S_{3}. In general, A_{n} ◁ S_{n}.
A group G is simple if it has no trivial, proper normal subgroups, e.g., ℤ_{p}, cyclic group of prime order, and A_{n} the alternating group, that is the group of even permutation of a finite set, for n ≥ 5.
A ring R is a non-empty set with two binary operations. It is an Abelian group under addition. A ring is associative, but not necessarily commutative under multiplication. When it is, we say that the ring is commutative. A ring does not necessarily have an identity under multiplication. When it does have an identity, it is called as unity or identity. Examples: ℤ, ℤ/ℤ_{n}.
A field F is a commutative ring with identity in which every nonzero element has an inverse or F^{x} = F -{0} is an Abelian group under multiplication. Examples: ℚ, ℝ, and ℂ are fields. $\mathbb{F_p}$ = ℤ/pℤ with p prime is a field. ℤ/ℤn is a field ↭ n is prime.
A subring A of a ring R is called a (two-sided) ideal of R if it absorbs multiplication from the left and right from R, that is, ∀r ∈ R, a ∈ A, ra ∈ A and ar ∈ A. ↭ ∀r ∈ R, rA = {ra | a ∈ A} ⊆ A and Ar = {ar | a ∈ A} ⊆ A ↭ ∀a, b ∈ A, ∀r ∈ R, a ∈ A, a - b ∈ A, ra ∈ A, ar ∈ A. Examples: nℤ is an ideal of ℤ.
A prime ideal I of a commutative ring R is a proper ideal of R such that a, b ∈ R, ab ∈ I ⇒ a ∈ I or b ∈ I ↭ R/I is an integral domain. Example: ⟨x⟩ ⊆ ℤ[x] is a prime ideal, but ⟨x^{2}⟩ is not, because x^{2} ∈ ⟨x^{2}⟩ but x ∉ ⟨x^{2}⟩.
A maximal ideal A of a commutative ring R is a proper ideal of R that is maximal with respect to set inclusion amongst all proper ideals, i.e., whenever B is an ideal of R, A ⊆ B ⊆ R ⇒ B = A or B = R. Let R be a commutative ring with unity and let I be an ideal of R. Then, R/I is a field iff I is maximal, e.g., ℤ[x]/⟨x⟩ ≋ ℤ is not a field, ⟨x⟩ is not a maximal ideal.
Let R be a commutative ring with unity. A principal ideal is an ideal generated by a single element a of R through multiplication by every element of R, ⟨a⟩ = Ra = {ra | r ∈ R} = {ar | r ∈ R}, e.g., ⟨x⟩ is a principal ideal of ℤ[x], but ⟨x, 3⟩ is not.
F field, p(x) ∈ F[x], ⟨p(x)⟩ is the set of all multiples of p(x), the principal ideal generated by p(x). F[x]/⟨p(x)⟩ is the quotient ring. If a(x) ∈ F[x], then a(x) + ⟨p(x)⟩ is the coset of ⟨p(x)⟩ represented by a(x). a(x) = b(x) (mod (p(x))) to mean that p(x) | a(x) - b(x).
Let R be a commutative ring with unity, let a_{1}, a_{2}, ···, a_{n} be a family of elements of R. Then, the ideal generated by the family a_{1}, a_{2}, ···, a_{n} is the set of element of R of the form r_{1}a_{1} + r_{2}a_{2} + ··· + r_{n}a_{n}, i.e., I = ⟨a_{1}, a_{2}, ···, a_{n}⟩ = ⟨r_{1}a_{1} + r_{2}a_{2} + ··· + r_{n}a_{n} | r_{i} ∈ R⟩
Let R be a commutative ring. A polynomial ring is a ring formed from the set of polynomials in one or more indeterminates or variables with coefficients in R. R[x] = {a_{n}x^{n} + a_{n-1}x^{n-1}+ ··· + a_{1}x + a_{0} | a_{i} ∈ R, n ∈ ℤ^{+}}, e.g. ^{2}⁄_{3}x^{2} + 4x -7 ∈ ℚ[x].
Let D be an integral domain. An irreducible polynomial is a polynomial that cannot be factored into the product of two non-constant polynomials. A non-zero and non-unit polynomial f(x) ∈ D[x] is said to be irreducible if it cannot be written as the product of two non-units. If f(x) = g(x)h(x), g(x) and h(x) ∈ D[x], then either g(x) or h(x) is a unit in D[x].
A non-constant polynomial f(x) ∈ F[x], F field, is said to be irreducible if f(x) cannot be expressed a as a product of two polynomials g(x), h(x) ∈ F[x] of lower degree, that is, f(x) = g(x)h(x), deg(g(x)) < deg(f(x)) and deg(h(x)) < deg(f(x)).
Definition. We say an integral domain D is a unique factorization domain, UFD for short, if:
If D is a unique factorization domain, then D[x] is a unique factorization domain. Examples: ℤ is a UFD (fundamental theorem of arithmetic) ⇒ ℤ[x] is a UFD. A field K is a UFD ⇒ K[x] is a UFD
A ring R is a principal ideal domain (PID) if it is an integral domain such that every ideal of R is a principal ideal, that is, has the form ⟨a⟩ = Ra = {ra | r ∈ R} = {ar | r ∈ R}, e.g., ℤ, and if F is a field, both F and F[x]. PID ⇒ UFD, but the reverse is not true.
Reducibility test for polynomials with degrees two or three. Let F be a field. If f(x) ∈ F[x] and deg(f(x)) = 2 or 3, then f(x) is reducible over F iff f has a zero in F.
Let D be an integral domain. The content of a non-zero polynomial f(x) = a_{n}x^{n} + a_{n-1}x^{n-1} + ··· + a_{0} ∈ D[x] is the greatest common divisor of its coefficients a_{n}, a_{n-1}, ··· a_{0}. A primitive polynomial is an element of D[x] with content 1, that is, gcd(a_{0}, a_{1}, ···, a_{n}) = 1, e.g., 3x^{2} -6x +9 is not primitive, 3x^{2} -6x +10 is primitive.
Gauss’s Lemma. Let D be a unique factorization domain. Let f(x), g(x) ∈ D[x]. Then, the content of f(x)g(x) equals the product of the content of f(x) and the content of g(x). In particular, if f(x), g(x) ∈ D[x] are primitive, then f(x)g(x) is also primitive.
Theorem. A non-constant polynomial in Z[x] is irreducible in Z[x] if and only if it is both irreducible in Q[x] and primitive in Z[x].
Eisenstein Criterion. p prime, f(x) = a_{n}x^{n} + a_{n-1}x^{n-1} + ··· + a^{0} ∈ ℤ[x], n ≥ 1, p | a_{0}, p | a_{1}, ···, p | a_{n-1}, but p ɫ a_{n}and p^{2} ɫ a_{0}. Then, f(x) is irreducible in ℚ[x].
For any prime p, the pth cyclotomic polynomial Φ_{p}(x) = $\frac{x^p-1}{x-1}=x^{p-1}+x^{p-2}+···+x+1$ is irreducible over ℚ.
Mod p Irreducibility Test. Let p be a prime, f(x) ∈ ℤ[x], and deg(f(x)) ≥ 1. Let f'(x) be the polynomial in ℤ_{p}[x] obtained from f(x) by reducing all the coefficients of f(x) modulo p. If f'(x) is irreducible over ℤ_{p} and deg(f'(x)) = deg(f(x)), then f(x) is irreducible over ℚ.
A field F is a commutative ring with identity in which every nonzero element has an inverse or F^{x} = F -{0} is an Abelian group under multiplication, e.g., ℚ, ℝ, ℂ. The characteristic of a field F, char(F), as the smallest positive integer p such that p·1_{F} = 1 + 1 + ··· + 1 (p times) = 0 if such a p exists, or char(F) = 0 if 1 + 1 + ··· + 1 ≠ 0 ∀n ∈ N, e.g., char(ℝ) = char(ℚ) = 0; and for any prime p, $\mathfrak{F}_p=ℤ/pℤ$ has characteristic p.
If F is a subfield of a field E, then E is called an extension field of F, denoted E/F or E:F. Let p(x) be an irreducible polynomial in F[x] and let u be a root of p(x) in an extension E of F. Then,
F(u), the subfield of E generated by F and u is the smallest field containing F and u. It is indeed F[u], the smallest subring containing both F and u = {b_{0} + b_{1}u + ··· + b_{m}u^{m} ∈ E | b_{0} + b_{1}x + ··· + b_{m}x^{m} ∈ F[x]}
The degree of p(x) is n, the set (1, u, ···, u^{n-1}) forms a basis of F(u) over F, i.e., each element of F(u) can be written uniquely as c_{0} + c_{1}u + ··· + c_{n}u^{n-1} where c_{i} ∈ F, and [F(u):F] = n.
An element α ∈ E is algebraic over F if there exists a non-constant polynomial p(x) ∈ F[x] such that p(α)=0, i.e., α is the root of p(x). Example: E = ℂ, F = ℚ, α = $\sqrt{2}$ is algebraic over ℚ, p(x) = x^{2} -2, and p($\sqrt{2}$)=0. Similarly, i is algebraic over ℚ, p(x) = x^{2} + 1, g(i) = 0. An element that is not algebraic, i.e., there no exists a non-constant polynomial p(x) such that p(α)=0 is called transcendental, e.g., π.
An extension E of F is called algebraic if each element of E is algebraic over F. An extension that is not algebraic, it is called transcendental extension, e.g, the extension ℝ over ℚ is a transcendental extension, and ℂ = ℝ(i) and ℚ($\sqrt{5}$) are algebraic extensions over ℝ and ℚ respectively.
Theorem. Let E be an extension field of F, and let u ∈ E be algebraic over F. Let p(x) ∈ F[x] be a polynomial of least degree such that p(u) = 0, we call p the minimal polynomial of u over E, Then,
(i) p(x) is irreducible over F
(ii) if g(x) ∈ F[x] is such that g(u) = 0, then p(x) | g(x)
(iii) There is exactly one monic polynomial p(x) ∈ F[x] of least degree such that p(u) = 0.
Let E be an extension of F,
If E is a finite extension of F, then E is an algebraic extension of F.
If p(x) is the minimal polynomial of α over F and degree(p(x))= n. Then, [F(α):F] = n, and in particular it is finite, so F(α) = F[α] is an algebraic extension. More formally, if E is an extension of F and α ∈ E is algebraic over F, then F(α) is an algebraic extension of F and deg(F(α):F)=deg(p(x)) where p is the minimal polynomial of α.
Multiplicative property of degree extensions. Let F ⊆ E ⊆ K be a tower of fields. If [K:E] < ∞ and [E:F] < ∞ then, (i) [K:F] < ∞, and (ii) [K:F] =[K:E][E:F]
A field homomorphism between two fields E and F is a function f: E → F such that ∀x, y ∈ E: f(x + y) = f(x) + f(y), f(xy) = f(x)f(y), f(1) = 1. It implies: f(0) = 0, f(x^{-1}) = f(x)^{-1}, and f is injective.
Let K/F and L/F be two extensions of F. An F-homomorphism of extensions of F is a field homomorphism σ: K → L such that σ(a) = a ∀a ∈F.
Let f(x) be a polynomial in F[x] of degree ≥ 1. Then, an extension K of F is called a splitting field of f(x) over F if:
Theorem. Existence of Splitting Fields. Let F be a field and let f(x) be a non-constant polynomial of F[x]. Then, there exists a splitting field E for f(x) over F,e.g., x^{2} + 1 ∈ ℚ[x] splits over ℂ and ℚ(i).
A finite field is a field that contains a finite number of elements,e.g., $\mathbb{F_p}=ℤ/pℤ$ The characteristic of a finite field have to be a prime number, say p. |F| = p^{n} for some n ∈ ℕ.
Structure theorem for finite fields. Let p be a prime, n be a positive integer, q = p^{n}. Then,
A field K is called algebraically closed if every non-constant polynomial f(x) ∈ K[x] has a root in K ↭ any irreducible polynomial of positive degree in K[x] has degree 1, e.g, ℂ is algebraically closed.
Theorem. Every field has an algebraic closure
Let F be a field and let p(x) ∈ F[x] be an 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) is isomorphic to F[x]/⟨p(x)⟩, F(a)≈F[x]/⟨p(x)⟩. Moreover, F(a) is a vector space over F with a basis {1, a, ···, a^{n-1}}, i.e., every member of F(a) can be written in the form c_{n-1}a^{n-1} + c_{n-2}a^{n-2} + ··· + c_{1}a + c_{0}, where c_{0}, c_{1}, ···, c_{n-1} ∈ F (Figure A)
Let ϕ: E → F be an isomorphism of fields. Let K be an extension field of E and α ∈ K be algebraic over E with minimal polynomial p(x). Suppose that L is an extension field of F such that β(∈L) is root of Φ(p(x)) in F[x]. Then, ϕ extends to a unique isomorphism $\barϕ :E(α)→F(β)$ such that $\barϕ(α)=β~ and~ \barϕ$ agrees with Φ on E (Figure B.)
Theorem. Let ϕ: E→ F be an isomorphism of fields and let p(x) be a non-constant polynomial in E[x] and q(x) the corresponding polynomial in F[x] under ϕ. If K and L are splitting field of p(x) and q(x) respectively, then ϕ extends to an isomorphism ψ: K → L (Figure C)
Theorem. Let F be a field, let E = F(α) be an extension of F, where α is algebraic over F and let f ∈ F[x] be the irreducible polynomial of α over F. Let Φ: F → K be a homomorphism from F to a field K, and let L be an extension of K. If β ∈ L is a root of Φ(f), then there is a unique extension of Φ to a homomorphism Φ: E → L such that ϕ(α) = β and fixes F.
Theorem. Let L/K be an algebraic extension. Every field homomorphism Φ: K → C where C is an algebraically closed field, can be extended to a homomorphism L → C
A multiplicative character, linear character, or simply character of a group G in a field F is a group homomorphism from G to the multiplicative group of a field, σ: G → F^{x} By homomorphism is meant a mapping σ such that ∀a, b ∈ G: σ(a)·σ(β) = σ(α·β).
The characters σ_{1},σ_{2}, ···, σ_{n} are independent if a_{1}σ_{1} + a_{2}σ_{2} + ··· + a_{n}σ_{n} ≡ 0 (≡ denotes equality as a function from G → F^{x}) for some a_{1}, a_{2},..., a_{n} ∈ F ⇒ a_{1} = a_{2} = ... = a_{n} = 0.
Theorem. If G is a group and σ_{1}, σ_{2}, ···, σ_{n} are n mutually distinct characters of G in a field F, then σ_{1}, σ_{2}, ···, σ_{n} are independent.
Independence of field homomorphisms. Let K, L be two fields, let σ_{1}, σ_{2}, ···, σ_{n}: K → L be distinct field homomorphisms (σ_{i}≠σ_{j}, ∀i, j: i≠j). If a_{1}σ_{1}(α) + a_{1}σ_{1}(α)+ ··· + a_{n}σ_{n}(α) = 0 ∀α ∈ K, then σ_{1}, σ_{2}, ···, σ_{n} are independent, i.e., a_{1} = a_{2} = ··· = a_{n} = 0.
Let K, L be two fields. If σ_{1},···, σ_{n}: K → L are mutually distinct field homomorphisms of K into L and let F be the fixed field of K, F = {a ∈ K| σ_{1}(a) = ··· = σ_{n}(a)}.
Let K be any field, and let σ_{1}, ···, σ_{n}: K → K be distinct field automorphisms. Suppose that σ_{1}, ···, σ_{n} forms a group under composition. If F is a fixed field of _{1}, ···, σ_{n}. Then, [K: F] = n ⇒ [K:F] = |G| = n.
The Galois group of the extension K/F, denoted by Gal(K/F), is the group of all F-automorphisms of K (the group of all automorphisms that fix the base field, Gal(K/F) = {σ ∈ Aut(K)| σ(a) = a ∀a ∈ F} e.g., $Gal(\mathbb{Q}(i)/\mathbb{Q})≋\mathbb{Z}/2\mathbb{Z},~ Gal(\mathbb{Q}(i, \sqrt{2})/\mathbb{Q})≋\mathbb{Z}/2\mathbb{Z} x \mathbb{Z}/2\mathbb{Z},~ Gal(\mathbb{Q}(\sqrt[3]{2})/\mathbb{Q})≋$ {e}, $Gal(\mathbb{F_{p^r}}/\mathbb{F_p})≋\mathbb{Z}/r\mathbb{Z}$
Degree of the fixed field extension is at least the number of homomorphisms. Let K, L be two fields. If σ_{1},···, σ_{n}: K → L are mutually distinct field homomorphisms of K into L and let F be the fixed field of K, F = {a ∈ K| σ_{1}(a) = ··· = σ_{n}(a)}. Then, [K : F] ≥ n.
Degree of Fixed Fields. Let K be any field, and let σ_{1}, ···, σ_{n}: K → K be distinct field automorphisms. Suppose that σ_{1}, ···, σ_{n} forms a group under composition. If F is a fixed field of σ_{1}, ···, σ_{n}. Then, [K: F] = n
Proposition. Let K be any field and let G be a finite group of automorphisms of K. Suppose F is the fixed field of G. Then G = Gal(K/F) = Gal(K/K^{G}).
To sum up. Let K be any field. S is any set of automorphisms of F. G is a group of automorphisms of K. [K : K^{S}] ≥ |S|, [K : K^{G}] = |G|, Gal(K, K^{G}) = G. F ⊆ K^{Gal(K/F)} ⊆ K. K/F is Galois if F = K^{Gal(K/F)}
Theorem. A finite extension K/F is Galois iff the order of the Galois group equals the degree of the extension, i.e., |Gal(K/F)| = [K : F]
Equivalence of Definition of Normal Extensions. Let K/F be a finite extension of fields. Let $\bar \mathbb{F}$ be an algebraic closure of F that contains K. Then, a field extension is called normal if it satisfies one of the following equivalent statements:
Let K be any extension of ℚ of degree 2. Then, we claim that Gal(K/Q) = ℤ/2ℤ, K/Q is normal and Galois.
Properties. If K is a normal extension of F, and L is an intermediate extension (that is, F ⊂ L ⊂ K) then K is normal extension of L. However, L/F is not necessarily normal.
Let K ⊆ L ⊆ F be finite extensions. Suppose K/F is a normal extension (⇒ K/L is normal). Then, L/F is a normal extension if and only if σ(L) ⊆ L ∀σ ∈ Gal(K/F).
Let F be any field.
Characterization of Galois extensions. Let K/F be a finite extension. Then, K/F is Galois ↭ K is the splitting field of a separable polynomial over F.
Theorem. Let K/F be a finite extension. The following statements are equivalent:
Fundamental Theorem of Galois Theorem. Let K/F be a finite Galois extension. Let G = Gal(K/F) be the Galois group. Then, the following statements holds, (1) There is an inclusion-reversing bijective correspondence between subgroups of G and intermediate fields of K/F, given by H → K^{H}, its inverse is defined by L → Gal(K/L). H_{1} ⊇ H_{2} ⇒ K^{H1} ⊆ K^{H2} and L_{1} ⊆ L_{2} ⇒ Gal(K/L_{1}) ⊇ Gal(K/L_{2}). Futhermore, it satisfies the following equality |H| = [K : K^{H}] and [G : H] = [K^{H} : F]
This map bijection fails when K/F is not Galois, e.g. K = $\mathbb{Q}(\sqrt[3]{2}), F = \mathbb{Q}$ is not Galois, G = Gal(K/F) = {1}. There is exactly one subgroup of G, but there are two intermediate fields, F and K.
(2) An intermediate fieldL (F ⊆ L ⊆ K) is Galois over F ↭ Gal(K/L) is a normal subgroup of G, Gal(K/L) ◁ G In this case, Gal(L/F) ≋ G/Gal(K/L)
Let K/F be a finite extension. An element α ∈ K is separable over F (A finite extension is always algebraic, and thus α is algebraic) if its irreducible polynomial over F is separable. A finite extension K/F is separable if every element of K is separable over F.
Definition. A field F is called perfect if either has characteristic zero or when F has characteristic p > 0, every element of F is a pth power.
Proposition. Let F be any field.
Theorem. Suppose f ∈ F[x] is irreducible. Then, f is separable if and only if f’ ≠ 0