# 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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### Field of Definition of an Algebraic Group

Linear Algebraic Groups- James E. Humphreys Chapter-XII Let $K$ be an algebraically closed field and $k$ be a arbitrary sub-field of $K.$ A closed set X in $A^n=K\times ...$(n times)$\times K$ is ...
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I asked this question: Characterization of the transcendentals over a field I realized I need some knowledge about transcendence degree to prove some facts in the book I'm reading. I would like to ...
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### Projective special linear groups are co-hopfian?

Is it known if $PSL(2,\, F)$, with $F$ a field of prime $p$ characteristic (maybe with all proper subfields of finite order), is co-hopfian? I've searched everywhere but found nothing. Definition A ...
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### Additive inverse in a field is unique

there will be an $a$ let's prove $-a$ is unique. let's assume there are 2 additive inverses $b$ and $c$ therefore $a+b=a+c$ let's multiply them by the multiplicative inverse of $a$ I try to use the ...
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### A Fixed Field of $\mathbb{Q}(\sqrt{2},\sqrt{3},\sqrt{5})$

I was working on a review problem from Dummit and Foote and came across the following issue. It is clear that the Galois group of the splitting field for the polynomial $(x^2-2)(x^2-3)(x^2-5)$ has ...
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### $F/K$ algebraic and every nonconstant polynomial in $K[X]$ has a root in $F$ implies $F$ is algebraically closed.

Let $F/K$ be an algebraic extension of fields in characteristic zero. If $F/K$ is normal, and every nonconstant polynomial $f \in K[X]$ has a root in $F$, then $F$ is algebraically closed. This is ...
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### Is there any subset of Complex numbers that is algebraically closed?

That any polynomial that is allowed to have coefficients from that subset has also a root in that subset
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### A question about algebraically closed fields

A field $\mathbb{K}$ is said to be algebraically closed in practice if every polynomial over $\mathbb{K}$ of positive degree less than or equal to $10^{10}$ has zero belonging $\mathbb{K}$. The ...
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### For extension fields, does $[F(a,b):F(a)]=[F(b):F]$?

sorry if this question seems obvious, For a field F and $a,b\notin F$, does $[F(a,b):F(a)]=[F(b):F]$? If so, how do you prove it or is there a counter example?
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### Is a finitely generated extension of a real closed field also real closed?

Let $\widetilde{\mathbb{Q}}$ be the field of real algebraic numbers, and consider $\widetilde{\mathbb{Q}}(\pi)$. My question: is $\widetilde{\mathbb{Q}}(\pi)$ a real closed field? Bonus karma points ...
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### A field having an automorphism of order 2

The following fact is used in the Unitary space. If $F$ is a field having an automorphism $\alpha$ of order 2. Let $F_0=\{a\in F: \alpha(a)=a\}$. Then $|F:F_0|=2$. Is there any easy proof (or ...
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### Show that these matrices are congruent.

Let $K$ be a field of characteristic$\ne 2$ and $u$ be an invertible element of $K$. Show that $\begin{pmatrix}1&0\\0&-1\end{pmatrix}$ and $\begin{pmatrix}u&0\\0&-u\end{pmatrix}$ are ...
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### Is it possible to describe $Q(x)$ as the extension field of $R$ freely generated by $\{x\}$?

Given an integral domain $R$, the polynomial ring $R[x]$ can be defined as the commutative $R$-algebra freely generated by $\{x\}$. Also, let $Q$ denote the field of fractions associated to $R$. Then ...
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### Galois closure of $\mathbb{C}(T,\sqrt{T^2+T+1})$ over $\mathbb{C}(T^3)$

I'm trying to solve the following problem, but it's too difficult for me. Let $\mathbb{C}(T)$ be the rational function field over the complex field $\mathbb C$, and put \$L:=\mathbb{C}(T,\sqrt{T^2 +...