Gauss's Lemma

Factorizing polynomials with rational coefficients can be difficult and Gauss's Lemma is a helpful tool for this problem. It is used implicitly in computer algebra packages.

Theorem A polynomial with integer coefficients that is irreducible in Z[x] is irreducible in Q[x]

Here's an example to illustrate the theorem. Consider f(x)=x³ - x² - x - 1. We are going to show that this polynomial is irreducible in Q[x]. Suppose that f(x) factorizes nontrivially in Z[x]. Since the highest common factor of its coefficients is 1 such a factorization has to be as g(x)h(x) where g and h both have degree <3. Thus

x³ - x² - x - 1 = (ax + b)(cx² + dx + e). Equating coefficients of x³ and x⁰ we get ac=1 and be=-1 so we see that a and b both have to be 1 or -1. It follows that f(x) has 1 or -1 as a zero but f(1) and f(-1) are both nonzero.

This contradiction shows that f(x) is irreducible in Z[x] and hence in Q[x]

Another application of the theorem is Eisenstein's irreducibility criterion.

We will prove a more general result than the theorem. First we need some definitions. We work with R[x] the polynomial ring over a unique factorization domain (UFD). Let f(x) in R[x], then we can find a highest common factor c, say, of the coefficients of f(x). Thus f=cg where g is in R[x] and is such that the highest common factor of its coefficient is 1. Such a polynomial is called primitive. The constant c is called the content of f. Note that c is only unique up to associates (so should really be called a content).

Gauss's Lemma Let R be a UFD and let f,g in R[x] be primitive. Then so is fg.

Proof Say f=f₀+f₁x+...+f_nxⁿ and g=g₀+g₁x+...+g_mx^m. Write fg=ch with h primitive and c the content of h. Suppose that p is a prime in R that divides c. Since f is primitive p cannot divide all its coefficients so choose f_i to be the first one not divisible by p. Likewise choose g_j to be the first coefficient of g not divisible by p. Now consider the coefficient of x^i+j in fg=ch. We have

ch_i+j = (f₀g_i+j + ... + f_ig_j +...+ f_i+jg₀).

Now because of the way f_i and g_j were chosen all the terms on the right hand side except f_ig_j are divisible by p. But p divides the right hand side by assumption. So we deduce that p|f_ig_j. This contradicts the primeness of p. We deduce that there is no such p and c is a unit. In other words, fg is primitive.

Corollary Let R be a UFD and let f,g in R[x]. Then content(fg)=content(f).content(g).

Proof: Write f=cf₁ and g=dg₁ where c,d are, respectively, the contents of f,g. Thus, fg=cdf₁g₁. Since f₁g₁ is primitive, by the Gauss Lemma, we are done.

Now for our UFD R let k= be the field of fractions of R. For example, if R=Z then k=Q. The next result has the very first theorem we stated as the special case when R=Z.

Theorem Let R be a UFD with field of fractions k. Let f be a polynomial in R[x]. Then f is primitive and irreducible in k[x] iff it is irreducible in R[x].

Proof: One direction is trivial. If f is primitive and irreducible in k[x] then how could it factorize over R[x]? By primitivity a factorization ch where c is a nonunit of R is impossible. By irreduciblity in k[x] such a factorization where both g,h have smaller degree than f is impossible.

So let's prove the other implications. Suppose that f is irreducible in R[x] (hence clearly primitive). Suppose that f=gh where g,h in k[x] have degree < deg f. clearing denominators we can rewrite this as

af=bg₁h₁ (*)

where a,b are in R and g₁,h₁ are in R[x]. Taking contents and using the corollary we get that a=b.content(g₁).content(h₁). Subsituting in (*) we have a nontrivial factorization of f in R[x], a contradiction.