Show that Z[root 7] is not isomorphic to Z[root 13].
How can i show this by contradiction?
We started a proof by contradiction, assuming there was an isomorphism
Well, consider you can see it is an element of whose square is (why?)
What we have to do now is to prove there is no squared root of in assume we already know that, then it's over because we obtained a contradiction.
To show the missing part, we can also use a proof by contradiction: suppose there is a such that
for some and we get since belongs to while is irrational or that means it is i.e.
being irrational, we can say and we finally have impossible: consider their squares and conclude.