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Recall. A unit or invertible element of a ring R is an element u ∈ R which has an inverse u^{-1} ∈ R, so that uu^{-1} = u^{-1}u = 1. An integral domain is a nontrivial commutative ring in which the product of any two nonzero elements is nonzero.
Definition. Let D be an integral domain. An irreducible polynomial is, roughly speaking, 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 or expressed as the product of two non-units. In other words, whenever f(x) is expressed as a product of two polynomials g(x)h(x) with 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)).
ℂ is algebraically closed, so all polynomials factors into linear terms.
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 or f(x) is irreducible over F iff f(α) ≠ 0 ∀α ∈ F.
Proof.
⇒) Suppose that f(x) is reducible ⇒∃g(x), h(x) ∈ F[x] such that f(x) = g(x)h(x). Since deg(f(x)) = deg(g(x)) + deg(h(x)) = 2 or 3 ⇒ At least one of g(x) and h(x) has degree 1 (2 = 1 + 1, 3 = 1 + 2). Without losing generality, we can assume ∃a∈ F, a ≠ 0_{F} such that g(x) = ax + b and f(x) = h(x)(ax +b) ⇒[a ≠ 0_{F}] -a^{-1}b is a zero of both g(x) and f(x).
⇐) Suppose that f has a zero in F, ∃α ∈ F: f(α) = 0 ⇒ [By the Factor theorem, f ∈ F[x], α ∈ F zero or root of f (f(α) = 0) ↭ (x -α)|f(x)] (x -α) is a factor of f(x), and therefore, f(x) is reducible over F ∎
x^{2} + 1 is irreducible over ℤ_{3}[x] (0^{2} + 1 = 1, 1^{2} + 1 = 2, and 2^{2} + 1 = 2), but reducible over ℤ_{5}[x] because 2 and 3 are roots (2^{2} + 1 = 3^{2} + 1 = 0), and therefore x^{2} + 1 = (x -2)(x -3) in ℤ_{5}[x].
In ℤ_{2}[x], f(x) = x^{3} +x^{2} + x + 1 is reducible because f(1) = 0, but g(x) = x^{3} + x + 1 is irreducible because g(1) = g(0) = 1.
Polynomials of degree larger than 3 may be reducible over a field, even though they do not have zeros, e.g., x^{4} + 2x^{2} + 1 = (x^{2} + 1)^{2} and x^{4} -5x^{2} + 6 = (x^{2} -2)(x^{2} -3) in ℚ[x], but they do not have any zeros in ℚ[x]. In ℤ_{2}[x], f(x) = x^{7} + x^{5} + x^{3} + x^{2} + 1 = (x^{4} + x + 1)(x^{3} + x + 1) is reducible even though f(0) = f(1) = 1.
f(x) = x^{3} +3x + 2 ∈ ℤ_{5}[x] is irreducible because deg(f(x)) = 3, f(α) ≠ 0 ∀α ∈ ℤ_{5}[x] (Figure 1.b.)
We know that if F is a field, then F[x] is a UFD, a unique factorization domain or factorial ring, that is, an integral domain (a nontrivial commutative ring in which the product of any two non-zero elements is non-zero) in which every non-zero non-unit element can be written as a product of prime or irreducible elements uniquely up to order and units.
An element p of a commutative ring R is said to be prime if it is not the zero element or a unit and whenever p divides ab for some a, b ∈ R ⇒ p divides a or p divides b.
Gauss’ lemma will say that if F is a field of fractions of R, then any polynomial f ∈ R[x] is in the UFD F[x], and so can be written as a product of irreducible factors in an essential unique manner. However, it produces factors of f in Irred(F[x]), but not necessarily in Irred(R[x]), and we only get uniqueness up to associates in F[x], which is a weaker result than uniqueness up to associates in R[x]. It turns out that the way to get around this problem is to consider primitive polynomials.
Definition. 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 equals to 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.
In particular, Gauss’s lemma over the integers states that if f(x) is a polynomial with integer coefficients, then F is called primitive if the greatest common divisor of all its coefficients is 1. If f and g are primitive polynomials over the integers, then their product fg is also primitive.
Proof.
Let f(x) = [We are basically factoring in its content, and everything left behind in f_{1} is relatively prime] cf_{1}(x) and g(x) = dg_{1}(x) where c and d are the content of f and g, respectively, and f_{1} and g_{1} are primitive polynomials.
Thus, f(x)g(x) = (cd)f_{1}(x)g_{1}(x). Therefore, it suffices to prove Gauss Lemma when f and g are primitive, and as a consequence, the content of fg would be equal to c·d·1. and we would have already proved the theorem for the general case.
Let f(x) = a_{n}x^{n} + a_{n-1}x^{n-1}+ ··· + a_{0} and g(x) = b_{m}x^{m} + b_{m-1}x^{m-1} + ··· + b_{0}. Suppose for the sake of contradiction, there’s a prime, say p, dividing all the coefficients of f(x)g(x). Since f and g are primitives, p does not divide all coefficients of f and g, so let r and s be the smallest integers such that p ɫ a_{r} and p ɫ b_{s}.
The coefficient of x^{r+s} in f(x)g(x) is given as,
c_{r+s} = $\sum_{i+j=r+s} a_ib_j = a_0b_{r+s}+a_1b_{r+s-1}+···a_rb_s+···+a_{r+s-1}b_1+a_{r+s}b_0$
Since p divides a_{0}, ···, a_{r-1} and b_{0}, ···, b_{s-1} ⇒ p | (c_{r+s} -a_{r}b_{s}) ⇒ [By assumption, p divides all coefficients of f(x)g(x), i.e. p | c_{r+s}] p | a_{r}b_{s} ⇒ [By Euclid’s lemma] p | a_{r} or p | b_{s} ⊥ (r and s be the smallest integers such that p ɫ a_{r} and p ɫ b_{s}.)
Therefore, the content of f(x)g(x) has no prime divisors in D, that is, the content is a unit.
Corollary. Let D be a UFD and F = Frac(D). Let f(x) ∈ D[x]. If there exists polynomials α, β ∈ F[x] such that f(x) = α(x)·β(x), then there exist polynomials a, b ∈ D[x] such that deg(α) = deg(a), deg(β) = deg(b), and f(x) = e·a(x)·b(x).
Notice that D[x] is in F[x] and f(x) = a(x)b(x) is a D-factorization. In particular, if f(x) ∈ ℤ[x] is reducible over ℚ, then it is reducible over ℤ. Example: f(x) = 2x^{5} + 5x^{4} -8x^{3} -14x^{2} +6x +9 ∈ ℤ[x], x = 3/2 is a root, therefore f(x) = (x -3/2)(2x^{4} +8x^{3} + 4x^{2} -8x + 6) = (2x -3)(x^{4} +4x^{3} + 2x^{2} -4x + 3), and this last one is a factorization over ℤ[x].
Proof:
Suppose that there exist polynomials α, β ∈ F[x] such that f(x) = α(x)β(x). Recall. Let D an integral domain ⇒ F the field of quotients of D, S = {(a, b)| a, b ∈ D and b ≠ 0}, (a, b) ~ (c, d) ↭ ad = bc, F = {[(a, b)]: (a, b) ∈ S}.
Let c, d ∈ D such that cα(x), dβ(x) ∈ D[x] where α, β are polynomials over the field of fractions (c and d are the least common multiple between all of the coefficients of the “denominators” of α and β respectively).
Next, we can factor out the greatest common factor of both polynomials, namely cα(x) = c_{1}a(x), cβ(x) = d_{1}b(x), where a(x), b(x) ∈ D[x] are primitive polynomials. We can see clearly that deg(a)=deg(α) and deg(b)=deg(β).
(cd)f(x) =[f(x) = α(x)β(x)] cα(x)·dβ(x) = c_{1}a(x)·d_{1}b(x) and by Gauss’ Lemma the content of c_{1}a(x)·d_{1}b(x) is c_{1}d_{1} (a(x), b(x) ∈ D[x] are primitive) ⇒ the content of (cd)f(x) is c_{1}d_{1} ⇒ cd | c_{1}d_{1} ⇒ say (cd)e = c_{1}d_{1} ⇒ [(cd)f(x) = (cd)ea(x)b(x), By cancellation] f(x) = ea(x)·b(x) is a factorization in D[x] and deg(a)=deg(α) and deg(b)=deg(β) ∎
Theorem. Let R be a unique factorization domain and F its field of fractions. A non-constant polynomial f in R[x] is irreducible in R[x] if and only if it is both irreducible in F[x] and primitive in R[x]. In particular, 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].
2x -4 and 3x + 3 are both irreducibles over ℚ, but 2x -4 = 2(x-2), and 3x + 3 = 3(x +1) are valid factorizations over ℤ (2 and 3 are not units in ℤ because 1/2 ∉ ℤ and 1/3 ∉ ℤ), and therefore 2x -4 and 3x + 3 are not irreducibles over ℤ. The problem is that 2x -4 and 3x + 3 are not primitives.
Theorem. D is a unique factorization domain if and only if D[x] is a unique factorization domain. Examples: ℤ is a UFD (fundamental theorem of arithmetic) ⇒ ℤ[x] is a UFD.
Proof.
⇐) D[x] is a UFD. All polynomials factors uniquely as a product of prime polynomials. In particular, the constant polynomials factors uniquely, and the constant polynomials are exactly the elements of the base domain D.
⇒) First, we need to prove the property of factorization. Let D be a UFD, then D is an integral domain and let F be its field of fractions. We know that F[x] is a Euclidean Domain, hence a PID, hence a UFD.
∀p(x) ∈ D[x], let’s view it inside F[x] ⇒ We can factor p(x), p(x) = f_{1}(x)··· f_{n}(x) with f_{i}(x) ∈ F[x] irreducible.
Let’s take a_{i} ∈ D such that a_{i}f_{i}(x) ∈ D[x] (basically we “clear” the denominators by multiplying these polynomials by their corresponding least common multiples of their denominators).
Next, we take b_{i} ∈ D such that a_{i}f_{i}(x) = b_{i}g_{i}(x) where g_{i}(x) ∈ D[x] are all both primitives (in a similar fashion as before by factoring out the greatest common factors of these polynomials coefficients) and irreducibles (f_{i} is irreducible and g_{i} is primitive ⇒ g_{i} is irreducible).
A non-constant polynomial f in R[x] is irreducible in R[x] if and only if it is both irreducible in F[x] and primitive in R[x]. a_{i}f_{i}(x) = b_{i}g_{i}(x) where g_{i}(x) ∈ D[x] are primitives, f_{i}(x) ∈ F[x] irreducibles ⇒ g_{i}(x) ∈ D[x] are irreducibles.
a_{1}···a_{n}p(x) = a_{1}f_{1}(x)···a_{n}f_{n}(x) = b_{1}g_{1}(x)···b_{n}g_{n}(x) = b_{1}···b_{n}g_{1}(x)···g_{n}(x) where g_{1}(x)···g_{n}(x) is primitive (g_{1}(x), g_{2}(x),···, g_{n}(x) are all primitives)
Set a = a_{1}···a_{n}, b = b_{1}···b_{n}, ap(x) = bg_{1}(x)···g_{n}(x) ⇒[g_{1}(x)···g_{n}(x) is primitive] a | b ⇒ ∃c ∈ D: b = ac ⇒ ap(x) = acg_{1}(x)···g_{n}(x) ⇒[By cancellation law, D is an integral domain] p(x) = cg_{1}(x)···g_{n}(x) ⇒ [D is a UFD, c ∈ D] p(x) = c_{1}···c_{m}g_{1}(x)···g_{n}(x) where g_{i}(x) ∈ D[x] are irreducibles.
It is left to the reader to prove the uniqueness of this factorization.