# Tagged Questions

Use this tag for questions about fields and field theory in abstract algebra. A field is, roughly speaking, an algebraic structure in which addition, subtraction, multiplication, and division of elements are well-defined. Please use (galois-theory) instead for questions specifically about that topic....

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### Why number of bases of $\mathbb{F}_p^2$ equals order of $GL_2(\mathbb{F}_p)$?

Artin, Algebra, Chapter 3, Ex. 4.4 I can prove (b), viz., that The order of $GL_2(\mathbb{F}_p)=p(p+1)(p-1)^2$ The order of $SL_2(\mathbb{F}_p)=p(p+1)(p-1)$ However, I have no idea how to prove (a)...
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### Field of the form $\{a+bi|a,b\in \mathbb{F}_p\}$

Artin, Algebra, Chapter 3, Ex 1.11 Consider whether the set of symbols $\{a+bi|a,b\in\mathbb{F}_p\}$ forms a field, if the laws of composition are made to mimic addition and multiplication of complex ...
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### The polynomial ring $K[t_1,\dots,t_n]$ of $n$ variables over the field $K$ has no zero divisors

Show that the polynomial ring $K[t_1,\dots,t_n]$ of $n$ variables over a field $K$ has no zero divisors (except the zero polynomial). When revising some Linear Algebra topics, I got stuck with this ...
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### Image of element not square of any element, maximal ideal, field is quadratic extension?

This is a followup to my question here. Say we have $\mathbb{F}_q$ a finite field, $\text{char.} \neq 2$, have $f \in \mathbb{F}_q[x]$, $f \notin \mathbb{F}_q$ be a squarefree element, and let us ...
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### What does algebraic mean?

If something is algebraic in a field, what does it mean? I don't know the correct phrasing. An element $a \in K$ is algebraic over $F$. Please can someone give some correct phrasing with a simple ...
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### Galois extension of intersection of fields

I have finite Galois extensions: E/K and E/L. $$M:=K \cap L$$ I am trying to prove that if the extension E/M is finite then it is also Galois. Any suggestions? Thanks
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### Calculating the fixed subfield of a splitting field $E$ corresponding to a subgroup of the Galois group $G = G(E/\mathbb{Q}$)

Here my splitting field is $E = \mathbb{Q}(\sqrt[3]{3}, \gamma)$, where $\gamma$ is a primitive cube-root of unity. This is the splitting field for $x^3-3$ in $\mathbb{Q}[x]$. I have calculated ...
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### $\sqrt[n]{x}$ as a power series in a complete field with absolute value.

Let K be a field complete with respect to a non-archimedean absolute value $| \cdot|:K^*\rightarrow \mathbb{R}$ and $char(K)=0$ or $char(K)\nmid n$. I want to prove that if $|x|\leq 1$ (i,e $x$ is in ...
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