# Tagged Questions

Galois theory allows one to reduce certain problems in field theory, especially those related to field extensions, to problems in group theory. For questions about field theory and not Galois theory, use the (field-theory) tag instead. For questions about abstractions of Galois theory, use (galois-...

50 views

### Find the Galois group of $x^4-x^2-6$.

I'm trying to find the Galois group of $x^4-x^2-6$. I think there are 4 roots, thus I guess the Galois group is $A_4$? But I don't know in general how to solve this.
42 views

57 views

51 views

### Showing normalizer of Galois group

Let $E/F$ be a Galois extension, and let $B$ be an intermediate field between $E$ and $F$. Let $H$ be the subgroup of $Gal(E/F)$ that maps $B$ into itself (but does not necessarily fix $B$). Prove ...
34 views

### Showing that the group of $n^{th}$ roots of unity form a group

I know they are $\{1, \gamma, \gamma^2, ..., \gamma^{n-1}\}$, where $\gamma$ is the primitive n-th root of unity. To show that this is a group (the four axioms), I did this: Since $\gamma^i$ is a ...
51 views

### for any $k \in \mathbb{Z}^{+}$, how do we know $\omega^1$ is a primitive root of $x^k - 1$?

I know that it is definitely a root. The set $\{1, \omega, ..., \omega^{k-1}\}$ are the roots of unity. But how are we guaranteed that it even exists and that furthermore it is a primitive root?
53 views

### How is $Gal(\mathbb{Q}(\sqrt[4]{3}, i)$ equal to $D_8$? (dihedral group of order 8)

From what I gathered, the automorphisms are as follows: $\iota$ (the identity map) $\sigma_2: \sqrt[4]{3} \rightarrow \sqrt[4]{9}$ (squaring map) $\sigma_3: \sqrt[4]{3} \rightarrow \sqrt[4]{27}$ (...
21 views

### Given a tower of field extensions, does this equality involving Galois group orders hold in general?

Suppose we have a tower of field extensions: $\overline{F} \subset K \subset E \subset F$ Is it true in general that $|G(K/F)| = |G(K/E)| \cdot |G(E/F)|$? I was able to verify some specific ...
28 views

### Showing that the multiplicative group of roots of $x^n - 1$ in $\overline{F}$ has $\phi(n)$ distinct generators

Basically the question is how can it be shown that the n-th cyclotomic extension, which is a cyclic group under multiplication, has exactly $\phi(n)$ distinct elements, where $\phi$ is Euler's totient ...
87 views

### Transitivity-like Results in Group, Ring, Module, Field and Galois Theory [closed]

I am reading Michael Atiyah and Ian Macdonald's Introduction to Commutative Algebra. On page 28, Proposition 2.16 says: Suppose $A,B$ are rings, $N$ is a finitely generated $B$-module, $B$ is ...
28 views

### polynomial in fixed field

Please suggest a method to solve this? Let $E$ be a field and let $H$ be a subgroup of $Aut(E)$. Let $F$ be the fixed field of $E$ under $H$ and let $f(X) ∈ E[X]$ be a monic polynomial that splits in ...
33 views

### Field extension if element fixed by only identity [closed]

Suppose that $E$ is a Galois extension of $F$ and that $α \in E$ is left fixed by only the identity in $\text{Gal}(E/F)$. Prove that $E = F (α)$. Please suggest how I should proceed. Thanks!
46 views

40 views

### Show that any group of order $2p$ is soluble

Let $|G| = 2p$. The result is clear is $p$ is even. The proof goes on to show that there is only one subgroup of order $p$ is p is an odd prime - my question is that, why do we need to show that ...
24 views

### Prove that $F \subset \sigma L$ is also radical.

Suppose that we have finite extensions $F \subset L \subset M$ and $\sigma \in Gal(M/F)$ and assume that $F \subset L$ is radical. Prove that $F \subset \sigma L$ is also radical. Since the ...
38 views

23 views

48 views

### Why is the Galois group for a cyclotomic extension isomorphic to $Z_n^{\times}$ and not to $Z_{n-1}$?

I'm struggling to understand this. Let $\mathbb{Q}(\xi)$ be the splitting field for $x^n - 1$, so here $\xi$ is the primitive n-th root of unity. If I consider the possible automorphisms of this ...
23 views

### Is $\prod_{\sigma\in \Sigma(L)}\mathbb Z$ free of rank $[K:\mathbb Q]$?

Let $L/\mathbb K$ be a Galois extension of number fields and $\Sigma(L)$ the set of embeddings of $L$ into $\mathbb C$. Let $H_L= \prod_{\sigma\in \Sigma(L)}\mathbb Z$. Let $G$ act on $H_L$ in the ...