Showing 2 Rings not Isomorphic

[jacobsldr]

The Question: Let Q[sqrt(2)] = {a + b(sqrt(2)), a,b elements of Q} and Q[sqrt(5)] = {a + b(sqrt(5)), a,b elements of Q} . Show that these 2 rings are not isomorphic. Q being the set of rationals.

Playing around with a direct map seems to be able to produce a simple ring homomorphism by preserving the addition and multiplication. By just staring at it for awhile, it sure seems like the direct map would be isomorphic, but I am wondering if there would be a problem with onto. I am trying to think what the image of sqrt 2 might look like. An irrational times a rational to be mapped to another irrational times a rational, not sure if there would be a problem there.

Any hints on this?? Thanks!

 

[daon]

Say \(\displaystyle \sqrt{2} \rightarrow a+b\sqrt{5}\)

Since f is a isomorphism, we immediately have \(\displaystyle a=0\).

Now, we have \(\displaystyle f(\sqrt{2})=b\sqrt{5}\).

Notice \(\displaystyle f(2)=f(\sqrt{2})f(\sqrt{2}) = 5b^2\)...

Now, what of \(\displaystyle f(1+1)\)?

 

[jacobsldr]

ahh...interesting. Ok, thanks very much!