# 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-...

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### The reducibility of polynomial $x^{2^n}+x+1$ over $Z_2$ [duplicate]

I meet the problem in Hungerford's Algebra, page 282 as follows: If $n>2$, then the polynomial $x^{2^n}+x+1$ is reducible over $Z_2$ . I have no any idea on this. But I really want to know how to ...
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### Intuition about structures in Galois Theory.

Background Often in mathematics I find we can ask "why" something is true. Of course, this is not a well defined question. However, it usually prompts answers that includes words like "intuitively..."...
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### Relation between divisibility of polynomials in different rings, $h | f$ in $\mathbb{Z}[x], \mathbb{Z}/p^k\mathbb{Z}[x]$ and $\mathbb{F}_p[x]$

Let $p$ be a prime, $k$ a positive integer. Let $f,h \in \mathbb{Z}[x]$ be polynomials such that $h | f \mod p^k$ in $(\mathbb{Z}/p^k\mathbb{Z})[x]$ $h \mod p$ is irreducible in $\mathbb{F}_p$ ...
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### Finding a primitive element for the Hilbert Class Field of $\mathbb{Q}(\sqrt{-14})$

I am trying to solve exercise 6.18 in the book of David Cox. I have tried to provide as much context as possible to make the situation as clear as possible for the reader. I have solved ...
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### What primes can ramify and decompose in $k(\mu_{p^m}) \mid k$?

Let $k$ be a number field and $p$ be a rational prime. Then consider the extension $k(\mu_{p^m}) \mid k$ of adjoining all roots of unity of degree $p^m$ to $k$. Assuming that $\mu_{p^m} \cap k$ is ...
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### How do we explain the existence of complex conjugation?

We can define the complex numbers by writing $\mathbb{C} = \mathbb{R}[i]/(i^2+1),$ where $\mathbb{R}$ is to be regarded as a commutative ring. Furthermore, since $\mathbb{R}$ happens to be a field, ...
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### Dummit and Foote on Galois and Representation Theory?

At some point, I'd like to learn both Galois Theory and Representation Theory. I currently know a lot of Group Theory and Linear Algebra, as well as some Ring Theory. I was thinking of reading ...
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### Square of normal covering splits

Concerning Galois theory, let $A/B$ be a separable extension. Then $$A/B - \text{normal} \Leftrightarrow A \otimes_B A=A \oplus \cdots \oplus A,$$ where the sum has $n$ summands. Is the same correct ...
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### Determine the fixed field of a subgroup of $S_3$ and check the Galois correspondence

I came across this homework problem and I'm stumped. This is the third part of the problem, so to give some context, here's what we have so far: Let $f(x)=x^3-2 \in \mathbb{Q}[x]$. We know that the ...
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### normal extension of $\mathbb{Q}$

We define an algebraic extension $K/F$ to be normal if every irreducible $f \in F[x]$ with one root in $K$ splits in $K[x]$. Now, in my lectures it was stated that $\mathbb{Q}(i)$ is a normal ...