# Tagged Questions

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### How to determine the Galois group of irreducible polynomials of degree $3,4,5$

Let $f$ be an irreducible (over $\mathbb{Q}$) polynomial in $\mathbb{Z}[x]$, $\deg (f)=3,4,5$. The Galois group of an irreducible polynomial $f\in \mathbb{Z}[x]$ acts transitively on distinct roots in ...
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### Values of virtual characters

Let $G$ be a finite group. Let $K$ be a number field and $K^c\subset\mathbb{C}$ its algebraic (separable) closure. Denote by $R_G$ the additive group of functions generated by the characters of ...
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### Possibilities for $[KL:F]$ when $[K:F]=[L:F]$ is prime

Suppose $K/F$ and $L/F$ are extensions of $F$ (contained in some common field) of degree $p$, where $p$ is prime. Standard arguments show that $[KL:F]$ must be in $\{p,2p,\ldots,p^2\}$. But are all ...
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### What could be said about a field extension, if it's Galois Group is fixpointfree?

Let $L/K$ be a galois extension, and let the Galoisgroup be fixpoint-free, what could be said about the field extension.
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### Is the Galois group associated to a random polynomial solvable with probability 0?

Choose a random polynomial $P\in\mathbb{Z}[x]$ of degree $n$ and coefficients $\leq n$ and $\geq-n$. Let $r_1,\ldots,r_n$ be the roots of $P$ and consider ...
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### What is the galois group of $x+3$ or $(x+1)(x+2)$ ? How about $A(x)B(x)$?

As the title says I wonder what the galois group of $x+3$ is. Or even if that exists ? Since $x+3 = 0$ has only one zero/element I assume its the trivial group ? And what is the galois group of ...
This is a question requires to compute the Galois group of $X^4+1$ over $\mathbb{Q}$, $\mathbb{Q}(i)$, $\mathbb{F}_3$ and $\mathbb{F}_5$. Here is a brief of what I can think of. For the first two, ...
### $p$-Sylow subgroups of a group of order $5^3\cdot 29^2$
I need to calculate the $p$-Sylow subgroups of a Galois group with order $5^3 \cdot 29^2$, i.e. $|\mathrm{Gal}(K/F)|=5^3 \cdot 29^2$. I've already established that there is only one 29-Sylow-subgroup ...