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

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manpreet Tuteehub forum best answer Best Answer 2 years ago

I am reading a syllaref="https://forum.tuteehub.com/tag/b">bus aref="https://forum.tuteehub.com/tag/b">bout Discrete Mathematics. One of the proref="https://forum.tuteehub.com/tag/b">blems encouraged in the syllaref="https://forum.tuteehub.com/tag/b">bus to solve is the following.

Define R={a+ref="https://forum.tuteehub.com/tag/b">b2:a,ref="https://forum.tuteehub.com/tag/b">bQ}R={a+ref="https://forum.tuteehub.com/tag/b">b2:a,ref="https://forum.tuteehub.com/tag/b">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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manpreet 2 years 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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