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Isomorphism between R={a+b2–√:a,b∈Q} and Q[x]/(f)

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Garry Buttler

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( 5 months ago )

I am reading a syllabus about Discrete Mathematics. One of the problems encouraged in the syllabus to solve is the following.

Define R={a+b2:a,bQ}R={a+b2:a,b∈Q} Find ff so ff defines an isomorphism between R and Q[x]/(f)Q[x]/(f). Any ideas on how to tackle this problem?

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Dilpreet Kaur

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( 5 months ago )

The idea is to consider the unique ring homomoprphism

φ:Q[x]Rφ:Q[x]→R
which satisfies φ(a)=aφ(a)=a for aQa∈Q, and φ(x)=2φ(x)=2.

 

Show this is surjective, and find that the kernel is the principal ideal generated by the minimal polynomial of 22 over QQ.

This assumes a bit of knowledge. It can be reformulated in more elementary terms, though. Please advise in case.

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