Proving Ring Isomorphism of Q[x]/<x^2-2> and Q[sqrt2]

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Homework Help Overview

The discussion revolves around demonstrating the ring isomorphism between Q[x]/ and Q[sqrt2], focusing on the structure of these mathematical entities.

Discussion Character

  • Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants explore hints for starting the proof, with one questioning the necessity of showing that x^2 - 2 is in the kernel. Another suggests directly writing down the isomorphism based on the forms of the elements in each structure.

Discussion Status

The conversation indicates that some participants are considering different approaches to establish the isomorphism, with one providing a potential function for the isomorphism. There is a recognition of the simplicity in the relationship between the two structures.

Contextual Notes

There is mention of a book example that may be overly detailed, suggesting that participants are navigating between formal proofs and intuitive understanding.

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what is the easiest way to show that
Q[x]/<x^2-2> is ring isomorphic to
Q[sqrt2]={a+b(sqrt2)|a,b in Q}

just give me a hint how to start
 
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anyone?

do I have to show that x^2 - 2 is in the kernal?
 
why don't you just write down the (obvious) isomorphism? (obvious in the sense of one side only has x as a special quantity, the other sqrt(2), and anything in Q[x]/(x^2-2) is of the form a+bx, isn't it...)
 
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yes! thank you. the example in the book goes into too much detail and I was trying follow that, but yes the function f(a+bx)=a+b(sqrt2) is a ring isomorphism.
 

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