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I need to prove if $\mathbb{Q}(\sqrt{3},i)$ is a Galois extension or not. I thought that $\mathbb{Q}(\sqrt{3},i)$ is the splitting field of $(x^2-3)(x^{2}+1)$ so it is separable, but I am not sure about it being normal; I think that it is. Could someone give some help?

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Hint: Look at the definition of a normal extension again. Every finite extension of $\Bbb{Q}$ is separable. – user38268 Feb 19 '13 at 2:11
Dear @Dimitri, I'm guessing your definition of Galois is separable and normal. As Gerry Myerson indicates in his answer, splitting fields are normal, but this is a non-trivial fact. The converse, however, that normal extensions are splitting fields, is immediate from the definitions. – Keenan Kidwell Feb 19 '13 at 2:14
Thanks, yes i know that splitting fields are normal, maybe i got confuse because i started with other descrition of the extension and then i see the fact that was the splitting field of a polynomial. Thanks for the answer to all. – Dimitri Feb 19 '13 at 2:22

If it's a splitting field, it's guaranteed to be normal. Since we are in characteristic zero, it's guaranteed to be separable.

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