Let $\displaystyle K = \mathbb {Q} [ \sqrt {m} ] $, where m is a square-free integer. Let $\displaystyle a=r+s \sqrt {m}$ be an algebraic integer in K.

a) Show that the minimum polynomial of a is $\displaystyle x^2-2rx+r^2-ms^2$

Now, I have verified that $\displaystyle (a)^2 - 2r(a)+r^2-ms^2 = 0 $

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 $\displaystyle [x-(r- \sqrt {m}s)][x+(r- \sqrt {m}s)] - 2rx$, will this help?

b) Show that $\displaystyle 2r, r^2-ms^2$ are integers.