Gauss' Lemma

agrotz2 16 Mar 2009 02:11

I came across two different definitions of Gauss' Lemma based upon which type of mathematics we look at. I

- In algebra, Gauss' lemma is either of two related statements about polynomials with integer coefficients.
- The first result states that the product of two primitive polynomials is primitive (a polynomial with integer coefficients is called primitive if the greatest common divisor of its coefficients is 1).
- The second result states that if a polynomial with integer coefficients is irreducible over the integers, then it is also irreducible if it is considered as a polynomial over the rationals.

- In number theory, Gauss' Lemma gives a condition for an integer to be a quadratic residue. Although it is not useful computationally, it has theoretical significance, being involved in some proofs of quadratic reciprocity.

I thought this was interesting since we had talked about defining things in a previous post. It just goes to show that definitions will differ depending on the application of the term/theorem.