Let , where m is a square-free integer. Let be an algebraic integer in K.
a) Show that the minimum polynomial of a is
Now, I have verified that
And I have to prove it is irreducible over K, should I use the norm, but the norm is not defined here yet, can I use N(ax+b) = a^2 + b^2?
I factored the polynomial into , will this help?
b) Show that are integers.
Okay, so then you should change to in your first post. Now, you are saying, "minimal polynomial" remember you need to specify over what field it is a minimal polynomial over, I assume it should be . In that case you have to prove that this polynomial has no rational zeros. Can you show that?
Since m is a square-free integer, is not a real number, so it is not a rational number. However, the only way to eliminate the m in the polynomial is to let , otherwise the m would remain. So there cannot be any rational zeros.
Is that sufficient to prove that ain't any rational zeros? One problem is I know it is true, I can explain it in words, but I don't know how to prove it mathematically.