I know that $\Bbb Q(\sqrt{3})$ is not field isomorphic to $\Bbb Q(\sqrt {5})$. What about group isomorphic and vector isomorphic? I think, it is group isomorphic under addition in both the sets. but how would we check vector space isomorphism ?
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3Any two vector spaces of the same finite dimension (over the same field) are isomorphic. – Ben Grossmann May 12 '16 at 18:16
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The additive group structure underlying a field is the additive group structure underlying the vector space structure underlying the field. – Travis Willse May 12 '16 at 18:23
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The answer is also given here. – Dietrich Burde May 12 '16 at 21:29
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Just do the map, you know
$$\begin{cases}\Bbb Q(\sqrt{3})=\operatorname{span}_{\Bbb Q}\{1,\sqrt 3\}\\ \Bbb Q(\sqrt{5})\cong \operatorname{span}_{\Bbb Q}\{1,\sqrt 5\}\end{cases}.$$
So map $a+b\sqrt 3\mapsto a+b\sqrt{5}$ and verify this is a linear bijection, i.e. an isomorphism of vector spaces (the inverse map is easy to define and gives the bijection, linearity is similarly straightforward)
Adam Hughes
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You can use the well-known fact that finite dimensional vector spaces with equal dimensions are isomorphic as vector spaces. Since linear isomorphisms are special cases of group homomorohisms, you know they are isomorphic as groups.
M. Van
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