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Math Help - Field problem with factor ring

  1. #1
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    Field problem with factor ring

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

    My proof so far:

    Now, 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 R[x] / <x^2+1>

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

    Let (ux+v+<x^2+1>) be in 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 \mathbb{C} just in case you are interested.

    Did you do the following theorems?

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

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

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