Questions tagged [extension-field]
Use this tag for questions about extension fields in abstract algebra. An extension field of a field K is just a field containing K as subfield, but interesting questions arise with them. Use this topic for dimension of extension fields, algebraic closure, algebraic/transcendental extensions, normal/separable extensions... This tag often goes along with the abstract-algebra and/or field-theory tags.
5,116 questions
1 vote
0 answers
42 views
Scheme over base field $k$ which is algebraically (resp. separably) closed in ring of regular functions of $X$
Let $X$ be a scheme over base field $K$ and denote by $R[X](= \Gamma(X,\mathcal{O}_X))$ the ring of regular global functions on $X$. Question: Assume that $k$ is algebraically (resp. separably) closed ...
- 10k
1 vote
0 answers
42 views
What can be said about $K((x))((y))\otimes_{K((x,y))} K((y))((x))$?
Let $K$ be a field. There is the iterated field of Laurent series $$ K((x))((y))=\{f:\mathbb{Z}^2\to K:f(x,y)=0\,\text{for}\,y<-N\,\text{or}\,y\ge -N,x<-N_y\}, $$ and similarly $K((y)((x))$. ...
- 2,783
8 votes
1 answer
234 views
Not all quadratic extensions over $\mathbb{Q}$ are contained in the compositum of all the splitting fields of irreducible cubics in $\mathbb{Q}[X]$
Question: Let $F$ be the composite of all the splitting fields of irreducible cubics over $\mathbb{Q}$. Prove that $F$ does not contain all quadratic extensions of $\mathbb{Q}$. (This is exercise 16 ...
- 813
2 votes
0 answers
42 views
How to choose $\alpha'$ for constructability of heptadecagon
I was going through this video: https://www.youtube.com/watch?v=UTxPV7zJ8EE If $z$ is the first 17th root of unity going counterclockwise from 1, define $\alpha=z_1+z^{-1}=2\cos(\frac{2\pi}{17})$ $\...
- 830
2 votes
1 answer
97 views
Basis for algebraic closure of $\mathbb{F}_2$
For any finite degree extension $\mathbb{F}_{2^t}$ of $\mathbb{F}_2$, we have at least a couple of canonical bases -- one being $\{1,\gamma,\ldots,\gamma^{t-1}\}$ where $\gamma$ is a primitive element,...
0 votes
1 answer
81 views
How to prove the simplest statement of Galois correspondence theorem in Galois extension?
Let $L/K$ is a finite field extension and assume it a Galois extension. I want to prove the following simplest form of the Fundamental theorem of Galois theory: $(a)$ For any subgroup $H$ of $\...
- 554
0 votes
1 answer
172 views
reciprocal of $\sqrt[3]{M}-\frac{1}{\sqrt[3]{M}}-a$ [duplicate]
Let $$ x=\sqrt[3]{M}-\frac{1}{\sqrt[3]{M}}+a $$ where $a\in\mathbb Q$ and $M$ is an integer such that $\sqrt[3]{M}\notin\mathbb Q$. Note that $x\in\mathbb Q(\sqrt[3]{M})$. Clearly $\frac{1}{x}\in\...
- 5,715
2 votes
2 answers
93 views
Bound on the degree over $k$ of a field extension $k\subseteq L \subseteq M_n(k)$
Let $k$ be a field, and $n$ be a positive integer. Then $k$ is embedded inside $M_n(k)$ (the set of $n×n$ matrices over $k$) as the set of scalar matrices $k'=\{aI_n:a \in k\}$. Suppose $L\subseteq ...
- 813