Date: Aug 9, 2012 10:54 AM
Subject: Very Short Abstract Algebra Question about Roots in Q[x]
Proposition: If a real number c is a root of an irreducible polynomial of degree strictly greater than 1 in Q[x], then c is irrational.
Proof [My Attempt]
We will prove by contradiction.
Let c be a real number that is a root of an irreducible polynomial of degree strictly greater than 1 in Q[x], call it p(x). And, to arrive a contradiction, suppose c is rational. But since c is a root of p(x) and c is rational, (x-c) is a factor of p(x) which is impossible since by our hypothesis asserts p(x) is irreducible. Thus, c must be rational. QED
Look correct? Thanks!