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.
When I plug in a into the poly, I have:
So it does equal to zero.
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.