1. ## 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]

2. Originally Posted by maroon_tiger
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?