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Thread: Field problem with factor ring

  1. #1
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    Field problem with factor ring

    Show that $\displaystyle R[x] / <x^2+1>$ is a field.

    My proof so far:

    Now, $\displaystyle R[x] / <x^2+1> = \{ ax+b+<x^2+1> : a,b \in R \}$

    Now, [tex](1+<x^2+1>)[tex] is the unity of $\displaystyle R[x] / <x^2+1>$

    But I have trouble trying to prove every element has a unit.

    Let $\displaystyle (ux+v+<x^2+1>)$ be in $\displaystyle R[x] / <x^2+1>$

    I need to find the inverse.

    little help, please?

    Thank you
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  2. #2
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    It happens to be isomorphic to $\displaystyle \mathbb{C}$ just in case you are interested.

    Did you do the following theorems?

    1.Given a commutative ring $\displaystyle R$ (with unity) and a maximal ideal $\displaystyle M$ then $\displaystyle R/M$ is a field.

    2.If $\displaystyle f(x)$ is a non-constant polynomial which is irreducible over the field then $\displaystyle \left< f(x) \right>$ is a maximal ideal of $\displaystyle F[x]$.

    So it remains to show that $\displaystyle x^2+1$ is irreducible over $\displaystyle \mathbb{R}$ which it is because it is a degree 2 polynomials with no real zeros.
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