Classify all the Galois group of the splitting field of $(x^2-2)(x^2-3)(x^2-5)$

Question

When the ground field is $\mathbb Q$ that is trivial. But what if the ground field has positive characteristic?

Is it still true that every Galois group has order 1,2,4,8?

Answer 1

If I understand correctly, you want to determine the galois group $G$ of the extension $L=K(\sqrt 2, \sqrt 3, \sqrt 5)$ over any field $K$. A priori $G \cong(\mathbf Z/2)^n$, with $n=0, 1, 2$ or $3$, and you wish to show that all values of $n$ can occur.

1) Suppose char $(K) \neq 2$. Since $K$ contains the group $(\pm 1)$ of square roots of $1$, Kummer's theory applies and tells you that $G\cong Hom (R, (\pm 1))$, where the so called Kummer radical $R$ is the subgroup of $K^*/{K^*}^2$ generated by the classes of $2, 3, 5$ mod ${K^*}^2$. A (multiplicative) linear dependence relation reads (@) $2^a.3^b.5^c =x^2$, with $x\in K^*$ and $a, b, c \in$ {0, 1}. Then : - if $K=\mathbf Q$, the relation (@) implies $a=b=c=0$ because $\mathbf Z$ is a UFD, so $n=3$ - take $K=\mathbf F_p$, where $p$ is an odd prime such that for instance $(\frac {5}{p})=-1$, so that $\mathbf F_p(\sqrt 5)/\mathbf F_p$ is a quadratic extension, actually the unique such extension in a fixed algebraic closure; then $L=\mathbf F_p(\sqrt 5)$ and $n=1$.

2) If char $K=2$, then $2=0$ , $3=5=1$, and $L=K$, i.e. $n=0$. See also complement (iii) below

3) If char $(K)$ is an odd prime $p$, see complement (iii) below

Complements. (i) If char$(K)=2$, when dealing with quadratic (not necessarily quadratic radical) extensions, one can apply Artin-Schreier's theory instead of Kummer's

(ii) Let $K=\mathbf F_q$, with $q$ = a power of an odd prime $p$. Since ${\mathbf F_q}^*$ is cyclic, it admits a unique subgroup of index $2$, which is no other than ${{\mathbf F_q}^*}^2$, and Kummer's theory tells us that necessarily $n=0,1$

(iii) If $K$ has odd characteristic $p$, radical quadratic extensions of the form $K(\sqrt a)$, with $a\in {\mathbf F_p}^*$, are classified by the image of the natural map ${\mathbf F_p}^*/{{\mathbf F_p}^*}^2\to K^*/{K^*}^2$, so in our case here, we still have $n=0,1$ over $K$

NB. Your question could also be interpreted as : for a fixed base field $K$, do all values of $n$ occur ? In characteristic $\neq 2$, the argument using Kummer's theory and @ still applies, but the answer obviously depends on $K$.