#How do I compute the ideal factorization R[X]/( 1+ x^2 ) ?

26 messages · Page 1 of 1 (latest)

tired shadow
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I know the answer is isomorphic to C but I want to see exactly how to do it

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marsh wraith
tired shadow
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G/Ker ~ Im?

marsh wraith
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The ring version since these are rings

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But yes

tired shadow
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Hmm

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Ok that proves the isomorphisms

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But how can I see the elements of R[X]/(1+x^2)?

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I mean how are the elements writen down explicitely

marsh wraith
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... they're remainders when you divide by 1+x^2

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And x maps down to a root of x^2+1

tired shadow
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eh

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sorry Im a bit rusty on that

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Can you write it mean for me

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the explicit isomorphism

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That way I can see it

marsh wraith
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f(x) maps to f(i)

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Trivially that is a surjection onto C

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It's evaluation so it's a ring homorphism

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And it's easy to prove the kernel is (x^2+1) since that's the minimal polynomial over R of i

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Oh sorry the iso

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Send the equivalence class of ax+b to ai+b

tired shadow
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Thanks

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+close