Show that Z[root 7] is not isomorphic to Z[root 13].

How can i show this by contradiction?

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- Mar 7th 2010, 07:16 AMBecksIsomorphic
Show that Z[root 7] is not isomorphic to Z[root 13].

How can i show this by contradiction? - Mar 7th 2010, 10:16 AMclic-clac
Hi

Assume there is a ring isomorphism

Using the morphism properties, try to find a contradiction with (Remember that, by definition, hence ) - Mar 7th 2010, 10:34 AMBecks
What are the morphism properties?

- Mar 7th 2010, 10:52 AMclic-clac
What appears in its definition, for instance, for any and

- Mar 8th 2010, 08:10 AMBecks
Could you start me off please?

- Mar 8th 2010, 11:13 AMclic-clac
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.