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In algebra, Gauss's lemma, named after Carl Friedrich Gauss, is a statement about polynomials over the integers, or, more generally, over a unique factorization domain (that is, a ring that has a unique factorization property similar to the fundamental theorem of arithmetic). Gauss's lemma underlies all the theory of factorization and greatest common divisors of such polynomials. Gauss's lemma asserts that the product of two primitive polynomials is primitive (a polynomial with integer coefficients is primitive if it has 1 as a greatest common divisor of its coefficients). A corollary of Gauss's lemma, sometimes also called Gauss's lemma, is that a primitive polynomial is irreducible over the integers if and only if it is irreducible over the rational numbers. More generally, a primitive polynomial has the same complete factorization over the integers and over the rational numbers. In the case of coefficients in a unique factorization domain R, "rational numbers" must be replaced by "field of fractions of R". This implies that, if R is either a field, the ring of integers, or a unique factorization domain, then every polynomial ring (in one or several indeterminates) over R is a unique factorization domain. Another consequence is that factorization and greatest common divisor computation of polynomials with integers or rational coefficients may be reduced to similar computations on integers and primitive polynomials. This is systematically used (explicitly or implicitly) in all implemented algorithms (see Polynomial greatest common divisor and Factorization of polynomials). Gauss's lemma, and all its consequences that do not involve the existence of a complete factorization remain true over any GCD domain (an integral domain over which greatest common divisors exist). In particular, a polynomial ring over a GCD domain is also a GCD domain. If one calls primitive a polynomial such that the coefficients generate the unit ideal, Gauss's lemma is true over every commutative ring. However, some care must be taken, when using this definition of primitive, as, over a unique factorization domain that is not a principal ideal domain, there are polynomials that are primitive in the above sense and not primitive in this new sense.
References
- Eisenbud, Exercise 3.4. (a)
- A generator of the principal ideal is a gcd of some generators of I (and it exists because [math]\displaystyle{ R }[/math] is a GCD domain).
- Atiyah & MacDonald, Ch. 1., Exercise 2. (iv)
- Atiyah & MacDonald, Ch. 1., Exercise 2. (iv) and Exercise 3.
- Atiyah & MacDonald, Ch. 1., Exercise 1.13.
- Eisenbud, Exercise 3.4.c; The case when R is a UFD.
- Proof for the GCD case: The proof here is adopted from Mines, R.; Richman, F.; Ruitenburg, W. (1988). A Course in Constructive Algebra. Universitext. Springer-Verlag. ISBN 0-387-96640-4. We need the following simple lemma about gcd: If [math]\displaystyle{ \gcd(a, b) = \gcd(a, c) = 1 }[/math], then [math]\displaystyle{ \gcd(a, bc) = 1 }[/math]. (The proof of the lemma is not trivial but is by elementary algebra.) We argue by induction on the sum of the numbers of the terms in [math]\displaystyle{ f, g }[/math]; that is, we assume the proposition has been established for any pair of polynomials with one less total number of the terms. Let [math]\displaystyle{ (c) = \gcd(\operatorname(fg)) }[/math]; i.e., [math]\displaystyle{ c }[/math] is the gcd of the coefficients of [math]\displaystyle{ fg }[/math]. Assume [math]\displaystyle{ (c) \ne (1) }[/math]; otherwise, we are done. Let [math]\displaystyle{ f_0, g_0 }[/math] denote the highest-degree terms of [math]\displaystyle{ f, g }[/math] in terms of lexicographical monomial ordering. Then [math]\displaystyle{ f_0g_0 }[/math] is precisely the leading term of [math]\displaystyle{ fg }[/math] and so [math]\displaystyle{ c }[/math] divides the (unique) coefficient of [math]\displaystyle{ f_0g_0 }[/math] (since it divides all the coefficients of [math]\displaystyle{ fg }[/math]). Now, if [math]\displaystyle{ c }[/math] does not have a common factor with the (unique) coefficient of [math]\displaystyle{ f_0 }[/math] and does not have a common factor with that of [math]\displaystyle{ g_0 }[/math], then, by the above lemma, [math]\displaystyle{ \gcd(c, \operatorname(f_0g_0)) = (1) }[/math]. But [math]\displaystyle{ c }[/math] divides the coefficient of [math]\displaystyle{ f_0g_0 }[/math]; so this is a contradiction. Thus, either [math]\displaystyle{ c }[/math] has a common factor with the coefficient of [math]\displaystyle{ f_0 }[/math] or does with that of [math]\displaystyle{ g_0 }[/math]; say, the former is the case. Let [math]\displaystyle{ (d) = \operatorname(c, \operatorname(f_0)) }[/math]. Since [math]\displaystyle{ d }[/math] divides the coefficients of [math]\displaystyle{ fg - f_0g = (f - f_0)g }[/math], by inductive hypothesis, [math]\displaystyle{ (d) \supset \operatorname(\operatorname((f - f_0)g)) = \operatorname(\operatorname(f - f_0)) \operatorname(\operatorname(g)) = \operatorname(\operatorname(f - f_0)) }[/math]. Since [math]\displaystyle{ (d) }[/math] contains [math]\displaystyle{ \operatorname(f_0) }[/math], it contains [math]\displaystyle{ \operatorname(f) }[/math]; i.e., [math]\displaystyle{ (d) = (1) }[/math], a contradiction. [math]\displaystyle{ \square }[/math]
- In other words, it says that a unique factorization domain is integrally closed.
