Note: This question comes from a non-examined question sheet from an undergrad maths course.
I want to find the splitting fields of the following polynomials:
- $x^3-1$ over $\mathbb{Q}$
- $x^3-2$ over $\mathbb{Q}$
- $x^3-2$ over $\mathbb{F}_5$
- $x^3-3$ over $\mathbb{F}_{13}$
In each case, I also want to show that the number of automorphisms of the splitting field is at most the degree of the extension.
Thoughts so far:
- The splitting field is $\mathbb{Q}(i, \sqrt{3})$, which has degree 4.
- The splitting field is $\mathbb{Q}(\sqrt[3]{3},e^{\frac{2 \pi i}{3}})$, which has degree 6.
- I note that $(x^3-2)$ has $3$ as a root in $\mathbb{F}_5$, and so $x^3-2=(x-3)(x^2+3x+4)$ in this field. We need a $\sqrt{5}$ to find the routes, but I'm not sure whether this can just be added to the field $\mathbb{F}_5$ in the same way that we add it to the field $\mathbb{Q}$. Help here would be appreciated.
- No thoughts on this yet other than to try each number from $1$ to $12$ and see if any work in $\mathbb{F}_{13}$ and to then try and reason as in (3).
Regarding the automorphisms, I would really like it if somehow could demonstrate how these calculations can be made, perhaps using one of the examples above as a template. I think I struggle conceptually and walk as methodologically with this part.
Any answers are much appreciated. Thanks.
Best Answer
General fact: automorphisms over the 'starting' field $F$ are as many as the degree of the extension over that field. Besides, if the extension is a splitting field, they are a group by composition (Galois group), a subgroup of the permutations of the roots. So, as an example, the first two polynomial discussion:
I suggest you to read again the theory of splitting fields and galois group: these are just basic concepts. If you need more examples, just ask for.