Can anyone help me to show isomorphism between $\mathbb Q(\sqrt{p})$ and $\mathbb Q(-\sqrt{q})$. where $p$ and $q$ are two distinct prime numbers. As i am check the property of isomorphism but I don't no how to prove it.
Please help, thank you
2026-03-25 15:48:01.1774453681
How to show isomorphism between extension fields?
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The two fields are not isomorphic because $\Bbb{Q}(\sqrt{p})$ contains a solution to $x^2-p=0$ whereas $\Bbb{Q}(-\sqrt{q})$ does not. This is easily verified by assuming towards a contradiction that there exist $a,b\in\Bbb{Q}$ such that $$p=(a+b\sqrt{q})^2.$$ Expanding the square quickly shows that no such $a,b\in\Bbb{Q}$ exist.