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Math Help - Quotient ring and field question

  1. #1
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    Quotient ring and field question

    For which prime numbers p is the quotient ring F_5[x]/(x^2+1) a field?

    From theorems in my notes, I know this quotient ring is a field if (x^2+1) is a maximal ideal, i.e (x^2+1) is irreducible modulo p over the field F_5[x].

    I don't see where to go from here though.. any help would be great
    Thank you!
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  2. #2
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    Quote Originally Posted by gizmo View Post
    For which prime numbers p is the quotient ring F_5[x]/(x^2+1) a field?

    From theorems in my notes, I know this quotient ring is a field if (x^2+1) is a maximal ideal, i.e (x^2+1) is irreducible modulo p over the field F_5[x].

    I don't see where to go from here though.. any help would be great
    Thank you!

    Twice you wrote \mathbb{F}_5[x] when it should, imo, be \mathbb{F}_p[x]

    Hint: -1 is a quadratic residue modulo p iff p=1\!\!\!\mod 4

    Tonio

    Ps. By the way, precisely in \mathbb{F}_5[x] \,,\,\,(x^2+1) is a not a maximal ideal.
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  3. #3
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    oops sorry! yes I do mean F_p[x]

    so, x^2=-1modp iff p=1mod4

    which is saying (x^2+1) is reducible iff  p=1mod4

    but we want to find p when (x^2+1) is irreducible:

    x^2 \neq -1modp or p \neq 1mod4

    I'm not really sure what to do from here though :/
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  4. #4
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    Quote Originally Posted by gizmo View Post
    oops sorry! yes I do mean F_p[x]

    so, x^2=-1modp iff p=1mod4

    which is saying (x^2+1) is reducible iff  p=1mod4

    but we want to find p when (x^2+1) is irreducible:

    x^2 \neq -1modp or p \neq 1mod4

    I'm not really sure what to do from here though :/

    You have it all above! First, a pol. of degree \leq 3 is reducible (over a given field) iff it has a root there; next, an ideal <f(x)> \leq \mathbb{F}[x]\,,\,\,\mathbb{F} a fiedl, is maximal iff f(x) is irreducible.

    So now you know exactly over what fields \mathbb{F}_p the polynomila x^2+1 is irreducible , so...

    Tonio
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  5. #5
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    so if p \neq mod4 ... p must satisfy one of these:

    p=0mod4 but this is not true as a prime number can't be divided by 4.
    p=2mod4 but this is also not true as it would mean p is even.

    Therefore, p=3mod4, in order for the quotient ring to be a field.

    I think that's it
    Thank you for your help! I would rep you if I could again
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