MathNugget
New member
- Joined
- Feb 1, 2024
- Messages
- 36
I am trying to find the Galois Group G([imath]\mathbb{Q(\sqrt{2}, \sqrt3)}/\mathbb{Q})[/imath]. That means, I have to find the automorphisms [imath]\phi: \mathbb{Q(\sqrt{2}, \sqrt3)} \rightarrow \mathbb{Q(\sqrt{2}, \sqrt3)}[/imath], for which [imath]\phi(x)=x, \forall x \in \mathbb{Q}[/imath].
Let's say K=[imath]\mathbb{Q(\sqrt{2}, \sqrt3)}[/imath].
I am fairly certain that all elements of K can be written as [imath]a + b\sqrt2 +c\sqrt3 + d\sqrt 6[/imath], [imath]a, b, c, d \in \mathbb{Q}[/imath].
I know that if [imath]K=\mathbb{Q}(\sqrt{c})[/imath], there are 2 automorphisms: [imath]a+b\sqrt{c} \rightarrow a-b\sqrt{c}[/imath] and identity. (let's say here that c is positive, and isn't the square of an integer or rational, so that [imath]\sqrt{c}[/imath] is irrational but not complex.
I also know that if [imath]K=\mathbb{R}[/imath] has just 1 automorphism satisfying this, the identity.
I have read the wikipedia page on Galois groups, there is a bit of info about it, but it doesn't satisfy my question. Would this field have 2^3 automorphisms? [imath]a \pm b\sqrt2 \pm c\sqrt3 \pm d\sqrt6[/imath]?
Let's say K=[imath]\mathbb{Q(\sqrt{2}, \sqrt3)}[/imath].
I am fairly certain that all elements of K can be written as [imath]a + b\sqrt2 +c\sqrt3 + d\sqrt 6[/imath], [imath]a, b, c, d \in \mathbb{Q}[/imath].
I know that if [imath]K=\mathbb{Q}(\sqrt{c})[/imath], there are 2 automorphisms: [imath]a+b\sqrt{c} \rightarrow a-b\sqrt{c}[/imath] and identity. (let's say here that c is positive, and isn't the square of an integer or rational, so that [imath]\sqrt{c}[/imath] is irrational but not complex.
I also know that if [imath]K=\mathbb{R}[/imath] has just 1 automorphism satisfying this, the identity.
I have read the wikipedia page on Galois groups, there is a bit of info about it, but it doesn't satisfy my question. Would this field have 2^3 automorphisms? [imath]a \pm b\sqrt2 \pm c\sqrt3 \pm d\sqrt6[/imath]?
