{"id":21,"date":"2016-09-22T19:01:48","date_gmt":"2016-09-22T19:01:48","guid":{"rendered":"http:\/\/keithmellinger.com\/magma\/?page_id=21"},"modified":"2016-09-22T20:18:28","modified_gmt":"2016-09-22T20:18:28","slug":"factoring-polynomials","status":"publish","type":"page","link":"https:\/\/keithmellinger.com\/magma\/factoring-polynomials\/","title":{"rendered":"Factoring Polynomials"},"content":{"rendered":"

We know that factoring a polynomial can be radically different depending on which field makes up your coefficient ring. Let\u2019s illustrate this using Magma<\/em>. We\u2019ll use the polynomial x5<\/sup> \u2013 5 and try to factor it over three different coefficient fields.<\/p>\n

To construct a polynomial ring over the ring of integers, we use the command:<\/p>\n

> P<x> := PolynomialRing(Integers());<\/code><\/p>\n

This assigns P to be the desired polynomial ring and the variable we\u2019ll use is x. Once you have the ring defined, you can create polynomials in that ring, and ask Magma <\/em>to factor them. Notice, however, that you must tell Magma<\/em> to think of the polynomial as \u201cliving\u201d in that specified ring. Polynomials such as x5<\/sup> \u2013 5 can live in many rings! Watch what happens as we change the coefficient ring:<\/p>\n

> P<x> := PolynomialRing(Integers());
\n> f := P! x^5 - x;
\n> Factorization(f);
\n[
\n  <x - 1, 1>,
\n  <x, 1>,
\n  <x + 1, 1>,
\n  <x^2 + 1, 1>
\n]
\n> P2<x> := PolynomialRing(GF(2));
\n> f := P2! x^5 - x;
\n> Factorization(f);
\n[
\n  <x, 1>,
\n  <x + 1, 4>
\n]
\n> P3<x> := PolynomialRing(GF(3));
\n> f := P3! x^5 - x;
\n> Factorization(f);
\n[
\n  <x, 1>,
\n  <x + 1, 1>,
\n  <x + 2, 1>,
\n  <x^2 + 1, 1>
\n]<\/code><\/p>\n

Exercise:<\/em><\/strong> There are many more commands in Magma<\/em> for working with polynomials. Define the polynomial ring P5 over the finite field with 5 elements. Then, use the IsIrreducible(f)<\/code> command to determine if the following polynomials are irreducible. If not, find the factorizations (over GF(5)).<\/p>\n