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Math Help - A couple questions about ideals

  1. #1
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    A couple questions about ideals

    I am trying to get some help on these problems

    1. alpha = 7 + sqrt(23), B = sqrt(alpha)
    Q: Find polynomials in I_{alpha, Q}
    {Note: to find these polynomials, they have to have 7+sqt(23) as a root. Do you have any advice on how to find a polynomial that has this as a root

    2. Prove that p in K[X] is irreducible if and only if <p(x)> is a maximal ideal in K[X]
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  2. #2
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    Quote Originally Posted by maroon_tiger View Post
    1. alpha = 7 + sqrt(23), B = sqrt(alpha)
    Q: Find polynomials in I_{alpha, Q}
    {Note: to find these polynomials, they have to have 7+sqt(23) as a root. Do you have any advice on how to find a polynomial that has this as a root
    What about (x-7+\sqrt{23})(x+7-\sqrt{23})? Multiply it out.

    2. Prove that p in K[X] is irreducible if and only if <p(x)> is a maximal ideal in K[X]
    Let p(x) be an irreducible polynomial. Form the ideal I = \left< p(x) \right>. Now suppose there is an ideal J such that I\subseteq J\subseteq K[x]. Since K[x] is a PID it means every ideal is principle, thus, J = \left< q(x) \right> . Now since I\subseteq J it means p(x) \in \left< q(x) \right>, thus, p(x) = q(x)h(x) for some h(x)\in K[x]. But since p(x) is irreducible it means either q(x) is constant in that case J=K[x] or h(x) is constant in that case J=I. Thus, I is a maximal ideal.
    Can you try the converse?
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