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ASADM

Member

Registered: 2009-03-05

Posts: 18

Permuting roots...

Hi guys,

This question is about Galois Theory.

One of the theorems in my books states that "Any K-automorphism of the field extension L/K must keep the elements of K invariant & must permute the roots of the minimal polynomial."

I was wondering, do the K-automorphisms permutes the roots of ALL polynomials, or just the minimal polynomial?

Many thanks in advance. :-) x

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Permuting roots...

I was wondering, do the K-automorphisms permutes the roots of ALL polynomials, or just the minimal polynomial?

All polynomials.  But remember, L is generated by the roots of a polynomial: L = K[a_1, ..., a_n].  So as soon as you define where these roots go, you define where everything in L goes.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."