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Topic: Very Short Abstract Algebra Question about Roots in Q[x]
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acadiaspring

Posts: 1
Registered: 8/9/12
Very Short Abstract Algebra Question about Roots in Q[x]
Posted: Aug 9, 2012 10:54 AM
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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!



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