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Consider a composed morphism $$\mathrm{Spec}(\mathbb{C}) \xrightarrow{f} \mathbb{A}^1_{\mathbb{C}} \xrightarrow{g} \mathbb{A}^1_{\bar{\mathbb{Q}}}$$ where $g$ is the natural base change morphism and $...
user45397's user avatar
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I think I may have a proof of the statement but I am told the statement is false, and can’t find an error in my proof, and am not sure how to construct a counter example. I have attached images of the ...
Pomegranate's user avatar
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Suppose $K$ is a number field and we have finitely many elements $\alpha_{1}, \ldots, \alpha_{k} \in K$ and a polynomial $f(x)\in K[x]$ of degree $\geq 2$. Given $m \geq 1$, we define $K_{m}(\alpha_{i}...
Gafar Maulik's user avatar
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1 answer
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Given a function field $K$ over $\Bbb C$ with a discrete valuation $v$, are there known criteria under which an automorphism of $K$ extends to the completion $K_v$? I’m aware of sufficient conditions ...
Anushka_Grace's user avatar
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11 votes
2 answers
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$\newcommand{\Z}{\mathbb Z}\newcommand{\F}{\mathbb F}$I would like to define a procedure which turns a finite $p$-group $G$ together with a field $K$ of characteristic $p$ (or even, in fact, any ring ...
Moinsdeuxcat's user avatar
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192 views

I would like to know what's the Hochschild cohomology algebra $$\operatorname{HH}^*(F,F)=\operatorname{Ext}^*_{F\otimes_kF}(F,F)$$ of the following field extension of a perfect field $k$, $$F=k(x_1,...
Fernando Muro's user avatar
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120 views

Let $G$ be a finite group, and $M$ be a minimal left ideal of $\mathbb{R}G$ (or irreducible $\mathbb{R}$-representation of $G$). There are three possibilities for $M$: Case 1: $M \otimes \mathbb{C}$ ...
khashayar's user avatar
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Let $(K,\bar{})$ be a field with a nontrivial involution whose fixed field is $K_0$. Does there exist a finite-dimensional simple $(K,\bar{})$-algebra $(A,*)$ with involution $*$ which is not central ...
khashayar's user avatar
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3 votes
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176 views

Let $F$ be a field, let $f \in F[x]$, let $E$ be the splitting field of $f$, and let $e \in E$ be written in terms of the roots of $f$. I'm looking for a simple way to establish if $e \in F$. If $E/F$ ...
en-drix's user avatar
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2 answers
290 views

I am currently reading Garrett's book "Holomorphic Hilbert Modular Forms". But I meet trouble at the starting line. Let $F = \mathbb{Q}(\sqrt D)$ be a real quadratic field, $u= a + b\sqrt{D}\...
Misaka 16559's user avatar
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Let $a$, $b$ be positive rational numbers such that $b$ is not the square of a rational number and $a^2-b$ is not a cube. Are these conditions sufficient to insure that the field ${\bf Q}(\sqrt[3]{a+\...
coudy's user avatar
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This question has been asked in Math.StackExchange (see here) for more than a week and I even put a bounty on it. But still it hasn't been correctly answered (the current answer there was written by ...
Z Wu's user avatar
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1 answer
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Let $L_1,\dotsc,L_n/K$ be finite separable field extensions. Then the compositum extension $L:=L_1\cdot\dotsb\cdot L_n/K$ is also finite and separable. Thus by the primitive element theorem, there are ...
Nicolas Banks's user avatar
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1 answer
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Let $k$ be a field of characteristic zero and $I$ be a radical ideal of $k[[x_1,\ldots, x_n]]$. Let $P$ be a minimal prime ideal of the reduced ring $R:=k[[x_1,\ldots, x_n]]/I$. Then, $R_P$ is a field ...
Alex's user avatar
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1 answer
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Let $u,v \in \mathbb{C}[x,y]$, where $u$ and $v$ are algebraically independent over $\mathbb{C}$ and $F=\mathbb{C}(u,v)$. Of course, $d:=[\mathbb{C}(x,y):F] < \infty$. Denote the following ...
user237522's user avatar
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Let $k$ be a field of characteristic zero and $I$ be a radical ideal of $k[[x_1,\ldots, x_n]]$. Let $P$ be a minimal prime ideal of the reduced ring $R:=k[[x_1,\ldots, x_n]]/I$. Then, $R_P$ is a field ...
Alex's user avatar
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For $r \geq 2$, let $A_r=\mathbb{C}[x_1,\ldots,x_r]$, $F_r=\mathbb{C}(x_1,\ldots,x_r)$ the field of fractions of $A_r$, and $E_r \subseteq F_r$ an arbitrary subfield of $F_r$ with $[F_r:E_r] < \...
user237522's user avatar
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Let $R \subseteq \mathbb{C}(x,y)$ and assume that $R=\mathbb{C}(u,v)$, where $u,v \in \mathbb{C}[x,y]$ are algebraically independent over $\mathbb{C}$. Here $\mathbb{N}$ includes $0$. Assume that $R$ ...
user237522's user avatar
  • 2,883
7 votes
0 answers
864 views

Let $K\subset L$ be a field extension and $A, B\subset L$ be $K$-subspaces of $L$ of finite positive dimensions. Assume further that for every $a, b \in L$ and every nontrivial proper finite ...
Shahab's user avatar
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2 votes
1 answer
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Motivation: Let me recall the well-known Artin-Schreier theorem (AST) for fields in a non-formal way; if $L$ is an algebraically closed field, and $K \subset L$ a subfield not 'much smaller' than $L$, ...
Maty Mangoo's user avatar
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1 answer
239 views

This is a question on algebraic geometry/commutative algebra. Let $K,L$ be fields of characteristics zero and let $K\subset L$ be a field extension (I am interested in the case when this is ...
S.J.'s user avatar
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6 votes
0 answers
300 views

I have changed some notation based on the comments of Chris Wuthrich and Wojowu. For an abelian extension $F$ of $\mathbb{Q}$, let $c(F)$ be its conductor. That is, $c(F)$ is the smallest positive ...
Steve Stahl's user avatar
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Let $k$ be an arbitrary base field and $K, L, M$ some fields over $k$ contained in a fixed overfield $\Omega$. Question: Are there some "reasonable" assumptions (ie beyond a bunch of really ...
user267839's user avatar
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103 views

Let $k$ and $\ell$ be division rings such that $\ell$ contains $k$, and $[\ell : k] = 2$. When do I know that there is an element $a \in k$ such that $x^2 = a$ has solutions in $\ell$, but not in $k$?
THC's user avatar
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For a rational number $r\in(0,1)$ the number $z=\sin(r\pi)$ is an algebraic number — such numbers appear to be called trigonometric numbers. What is its degree? That is, what is the minimal degree of ...
Joonas Ilmavirta's user avatar
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1 vote
0 answers
104 views

Let $f(X) \in k[T]$ be irreducible over the field $k$, and separable of finite degree $n$. Then if $\ell$ is the corresponding field extension, we know by Galois theory that $\mathrm{Gal}(\ell/k)$ ...
THC's user avatar
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0 votes
1 answer
528 views

In proving Monsky's Theorem, it is required that we define the $2$-adic absolute value on an arbitrary finitely generated extension of $\mathbb{Q}$ say $\mathbb{K}=\mathbb{Q}(\alpha_1,\ldots,\alpha_n)$...
user avatar
4 votes
1 answer
299 views

Let $p$ be a prime, and let $K/\mathbb{Q}_p$ be a tamely ramified finite extension of degree $n$. Let $q$ be a prime factor of $n$ with $q\neq p$. Must there exist an intermediate extension $L$ (...
Ralph Morrison's user avatar
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2 votes
0 answers
135 views

Any connection between extension of algebraic structure and forcing of set theory? And more, are there any approach from one of the two to other field to solve problem?
XL _At_Here_There's user avatar
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2 votes
1 answer
549 views

$\DeclareMathOperator\char{char}$This question is inspired by the MSE question Example of a non-algebraically closed field without quadratic extensions. To repeat: Given $n\in\mathbb{N}_{>1}$ ...
Thomas Preu's user avatar
  • 231
6 votes
1 answer
398 views

Let $f(x)\in K((x))$ be an algebraic formal Laurent series and let $P(x,y)\in K(x)[y]$ be its minimal polynomial. Is $P(x,y)$ always separable? An example of non separable polynomial comes from ...
Jiu's user avatar
  • 385
2 votes
0 answers
180 views

$\DeclareMathOperator\Imm{Im}$Let $K=F(\alpha)$ be an abelian extension of $F$ and let $\sigma$ be a map (could be any map) from $K^\times$ (the multiplicative group of $K$) to itself. Define an $F$-...
user44312's user avatar
  • 643
5 votes
1 answer
873 views

Let $A$ be a finite-dimensional algebra over a field $F$. A representation $M$ of $A$ is called absolutely irreducible if $M\otimes_FE$ is irreducible as a representation of $A\otimes_FE$ for all ...
Hebe's user avatar
  • 961
1 vote
0 answers
155 views

Let $K=\mathbb{C}(t_1,\dots,t_n)$ be the field of rational functions, $f$ an algebraic function over $K$ and assume the field extension $K(f)/K$ is non-solvable. Is it possible to characterise the ...
12345's user avatar
  • 171
2 votes
0 answers
150 views

Let's consider a set of numbers that one knows to high precision, and one knows or has a strong suspicion that `exact versions of these numbers' (see below) belong to a real extension of $\mathbb{Q}$. ...
eddy ardonne's user avatar
  • 603
2 votes
2 answers
610 views

I posed this question on Math.Stackexchange (see here) but until now there was no response. This made me decide to give it a try here. Let $k\subseteq F$ denote an algebraic field extension and let $\...
drhab's user avatar
  • 217
3 votes
2 answers
291 views

Let $\Omega$ be an algebraically closed field of characteristic $0$, $k$ a subfield such that $\mathrm{tr.deg}(\Omega/k)=\infty$. Let $u_1,\dots,u_n,u_{n+1}\in \Omega$ be algebraically independent ...
Makimura's user avatar
  • 113
3 votes
0 answers
242 views

Given an algebraically closed complete valued field $(k,|.|)$ with characteristic 0, such that the residue field $\tilde{k}$ has a positive characteristic, and consider the complete extension $(\...
AZZOUZ Tinhinane Amina's user avatar
  • 273
6 votes
0 answers
274 views

I asked this question over a year ago on Math.StackExchange but I didn't get an answer. In his famous treatise On spirals, Archimedes used a spiral to square the circle and trisect an angle. There are ...
J.-E. Pin's user avatar
  • 851
1 vote
2 answers
471 views

Let $F\subset E$ be a field extension. So $E$ has a natural structure of $F$-vector space. A vector subspace $V\subset E$ is a special subspace if $F\subset V$ and $V$ is closed under the inverse ...
Ali Taghavi's user avatar
  • 400
1 vote
1 answer
132 views

I need to find "Algorithms for Polynomials Over a Real Algebraic Number Field Ph.D. thesis, University of Wisconsin, Madison (1974) by Rubald". However I cannot find it online nor in my ...
Lucio Tanzini's user avatar
  • 457
8 votes
3 answers
2k views

Most proofs of Galois theorem stating that "an equation is solvable in radicals if and only if its Galois group is solvable," show the left to right direction by induction on the height of ...
Cyril's user avatar
  • 221
1 vote
1 answer
254 views

I need a reference for field extension and Galois extension (like an introduction) please. Thank you.
Tohiea's user avatar
  • 131
2 votes
0 answers
458 views

Let $K/L/M$ be a tower of finite field extensions with $K/L$ separable and $L/M$ simple (in the sense of being generated by a single element). How does one show that $K/M$ is also simple? I know that ...
One More Question's user avatar
  • 61
6 votes
2 answers
858 views

If $K$ is a finitely generated field extension of $k$, then there exists an irreducible affine $k$-variety with function field $K$. The idea is that if $x_1, \dots, x_n$ are generators of $K$ under $k$...
Federico Fallucca's user avatar
  • 305
9 votes
1 answer
364 views

The following is a known result in algebraic geometry: Let $k$ be an algebraically closed field of characteristic zero (for example, $k=\mathbb{C}$). Let $L$ be a field such that $k \subset L \subset ...
user237522's user avatar
  • 2,883
2 votes
2 answers
268 views

Let $q$ be a prime power. Let $\mathbb{F}_q$ be the finite field with $q$ elements. Then $\mathbb{F}_{q^n}$ is a field extension of $\mathbb{F}_q$ of degree $n$ and can be considered as an $n$-...
Daps's user avatar
  • 550
3 votes
0 answers
172 views

If $E/F$ is algebraic extension of finite degree $n$, then if $\alpha \in E$ is an element, then the degree of minial polynomial $m_\alpha$ for $\alpha$ is at most $n$. Even better, $\deg m_\alpha$ ...
Michal Dvořák's user avatar
  • 131
11 votes
3 answers
1k views

For a cubic polynomial $f(x)=x^3+x^2+\frac{1}{4}x+c$ over $\mathbb{F}_q$, where $q$ is a odd prime power, I find that for a lot of $q$, there does not exist $c\in\mathbb{F}_q$ such that $f$ has three ...
user avatar
6 votes
2 answers
568 views

Let $\pi$ be a representation of a Lie algebra $L$ in a finite-dimensional linear space $V$ over the field $F$. Let $K$ be a field extension of $F$. Let $\pi_K=\pi\otimes K$ be the corresponding ...
liorz's user avatar
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