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I want to know how to find a Galois group. I know I need to look at automorphisms but I am not sure of the method. Could someone give me an idea, e.g. with the example: $\mathrm{Gal}(\mathbb Q(\sqrt 2,\sqrt3,\sqrt5)/\mathbb Q$ Thanks

Jyrki Lahtonen
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thinker
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    Your question is probably too broad. There are many techniques for computing Galois groups, e.g., for showing that the problem is algorithmic under certain hypotheses. See, for example, chapter 13 of Cox's Galois Theory. – Gro-Tsen Jan 19 '16 at 18:05
  • Also, it turns out that $\mathbb{Q}(\xi)=\mathbb{Q}(\sqrt{2},\sqrt{3},\sqrt{5})$ where $\xi=\sqrt{2}+\sqrt{3}+\sqrt{5}$, so it is Galois with group $(\mathbb{Z}/2\mathbb{Z})^3$ (changing each root's sign independently), but this is far from obvious without a computer: $\sqrt{2}=(\xi^7-28\xi^5-56\xi^3+960\xi)/576$, for example, something which is rather a pain to check by hand. – Gro-Tsen Jan 19 '16 at 18:11
  • @Gro-Tsen sorry i made a mistake, i didnt mean to put '+' signs , i meant ',' . Does it look more solvable now – thinker Jan 19 '16 at 18:21
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    Yes, I also answered that in passing: it turns out (though it is by no means obvious!) that the field is the same with $+$ as with $,$, and the Galois group is $(\mathbb{Z}/2\mathbb{Z})^3$, essentially because changing the sign of any one of $\sqrt{2},\sqrt{3},\sqrt{5}$ independently of the others gives a field automorphism. I don't have the time to elaborate right now, probably someone else will. But I'll give more interesting/difficult examples to think about in the next comment. – Gro-Tsen Jan 19 '16 at 18:25
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    Here are three examples of Galois groups one can compute by hand and which I think are very instructive (especially interesting is the difference between the three): (a)$\mathbb{Q}(\sqrt{2\pm\sqrt{5}}$, (b)$\mathbb{Q}(\sqrt{2\pm\sqrt{2}}$ and (c)$\mathbb{Q}(\sqrt{2\pm\sqrt{3}}$, all over $\mathbb{Q}$. Answers are: (a)the dihedral group of the square, (b)the cyclic group $\mathbb{Z}/4\mathbb{Z}$ with four elements, and (c)$(\mathbb{Z}/2\mathbb{Z})^2$. – Gro-Tsen Jan 19 '16 at 18:30
  • Thanks @Gro-Tsen what is the procedure for finding these galois groups? – thinker Jan 19 '16 at 19:32
  • I think this recent thread as well the older threads linked to in that question should help you quite a bit. – Jyrki Lahtonen Jan 19 '16 at 21:43
  • @thinker: Related: http://math.stackexchange.com/questions/1367383/determine-the-galois-group-of-mathbbq-sqrtab-sqrtd, http://math.stackexchange.com/questions/1634569/, http://math.stackexchange.com/questions/575171/, http://math.stackexchange.com/questions/256574 (and http://math.stackexchange.com/questions/1215186, http://math.stackexchange.com/questions/86259, http://math.stackexchange.com/questions/1241221) – Watson Aug 29 '16 at 20:46

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