Questions tagged [field-extensions]
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56 questions
2 votes
1 answer
142 views
Are there known necessary conditions for when a function field automorphism extends to an automorphism of its completion?
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 ...
- 137
11 votes
2 answers
548 views
Extending scalars of p-groups
$\newcommand{\Z}{\mathbb Z}\newcommand{\F}{\mathbb F}$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 ...
- 535
4 votes
0 answers
184 views
What is the Hochschild cohomology algebra of a function field of infinite transcendence degree?
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,...
- 15.7k
3 votes
0 answers
117 views
primitive idempotents in semisimple group algebras
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: $M \otimes \mathbb{C}$ ...
- 203
1 vote
1 answer
118 views
Fixed field of involutions in simple algebras
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 ...
- 203
3 votes
1 answer
173 views
Is there a (simple) criterion for membership to the base field of an inseparable extension?
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 $E/F$ ...
- 424
0 votes
2 answers
286 views
What is the definition of Tr in the context of Hilbert modular forms?
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}\...
- 315
4 votes
1 answer
577 views
Non-trivial subfield of ${\bf Q}(\sqrt[3]{a+\sqrt{b}})$
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+\...
- 20k
10 votes
1 answer
325 views
If $E_\text{sep}/F$ is normal, then must $E/F$ be normal?
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 ...
- 632
4 votes
1 answer
247 views
Defining polynomial of compositum of splitting fields
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 ...
- 173
1 vote
1 answer
160 views
On analytic transcendence degree and Krull dimension for homomorphic images of power series rings
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 ...
- 480
1 vote
1 answer
212 views
$F=\mathbb{C}(u,v)$ satisfying: For every $a,b \in \mathbb{C}[y],c,d \in \mathbb{C}[x]$: $\mathbb{C}(x,y)=F(ax+b)=F(cy+d)$
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 ...
- 2,875
1 vote
1 answer
152 views
Transcendence degree and Krull dimension for homomorphic images of power series rings
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 ...
- 480
1 vote
0 answers
89 views
If $E \subseteq F=k(x_1,\ldots,x_r)$, satisfies $E(x_1^{i_1},\ldots,x_r^{i_r})=F$, for every $(i_1,\ldots,i_r) \neq (0,\ldots,0)$, then $[F:E] \leq 2$
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] < \...
- 2,875
0 votes
1 answer
574 views
A subfield $R \subseteq \mathbb{C}(x,y)$ with 'many' generators $w$, $R(w)=\mathbb{C}(x,y)$
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 $R$ ...
- 2,875
7 votes
0 answers
857 views
Dimension inequality for subspaces in field extensions
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 ...
- 379
2 votes
1 answer
296 views
Artin-Schreier theorem for rings (a little different)
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 $L$, ...
- 778
1 vote
1 answer
239 views
Shrinking the base field of an affine variety
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 ...
- 39
6 votes
0 answers
295 views
Abelian extensions of Q and cyclotomic fields
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 ...
- 395
0 votes
0 answers
174 views
Identity for compositum and intersection of fields
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 ...
- 3,814
0 votes
0 answers
102 views
Squares in division ring extensions $\ell/k$ with $[\ell:k] = 2$
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 $k$?
- 4,781
1 vote
0 answers
169 views
Degrees of trigonometric numbers
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 ...
- 8,166
1 vote
0 answers
103 views
Galois action on blow-ups related to field extensions of infinite degree
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 $\mathrm{Gal}(\ell/k)$ ...
- 4,781
0 votes
1 answer
522 views
How do I extend the $2$-adic absolute value to prove Monsky's Theorem?
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 $\mathbb{K}=\mathbb{Q}(\alpha_1,\ldots,\alpha_n)$...
user500557
4 votes
1 answer
295 views
Existence of intermediate field extensions for tamely ramified p-adic extensions
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 $L$ (...
- 421
2 votes
0 answers
134 views
Any connection between extension of algebraic structure and forcing of set theory?
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?
- 3,296
2 votes
1 answer
515 views
Fields with restrictions on their finite extensions: Given $n\in\mathbb{N}_{>1}$ which fields $F$ do not have extensions of degree $n$?
$\DeclareMathOperator\char{char}$This question is inspired by the MSE question Example of a non-algebraically closed field without quadratic extensions. To repeat: Given $n\in\mathbb{N}_{>1}$ ...
- 231
6 votes
1 answer
395 views
Is the minimal polynomial of an algebraic formal Laurent series always separable?
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 ...
- 385
2 votes
0 answers
180 views
A dimension problem related to an abelian simple extension of a field
$\DeclareMathOperator\Imm{Im}$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 $F$-...
- 643
5 votes
1 answer
857 views
Absolutely irreducible representation and splitting field
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 ...
- 961
1 vote
0 answers
149 views
Indeterminacy locus of an algebraic function
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 ...
- 171
2 votes
0 answers
150 views
Finding elements in a real extension of $\mathbb{Q}$ that are close to some number in $\mathbb{R}$
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 $\mathbb{Q}$. ...
- 603
2 votes
2 answers
594 views
If $G=\mathsf{Aut}_k(F)$ acts on field $F$ algebraic over $k$ then do we have: orbit $G\alpha=\text{ roots of minimal polynomial of }\alpha$?
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 $\...
- 217
3 votes
2 answers
278 views
Existence of generic zeros
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 ...
- 113
3 votes
0 answers
226 views
Wildly ramified extension field
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 $(\...
6 votes
0 answers
264 views
The power of Archimedean spirals: is there an algebraic characterization of Archimedean numbers?
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 ...
- 851
1 vote
2 answers
467 views
A quantity associated to a field extension
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 ...
- 226
1 vote
1 answer
128 views
Algorithms for Polynomials Over a Real Algebraic Number Field, a reference
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 ...
- 457
8 votes
3 answers
2k views
Counter example of a radical extension that is not Galois/normal over $\mathbb{Q}(\omega)$?
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 ...
- 221
1 vote
1 answer
253 views
Reference book for Galois extension [closed]
I need a reference for field extension and Galois extension (like an introduction) please. Thank you.
- 131
2 votes
0 answers
445 views
A composition of a simple extension and a separable extension is simple
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 ...
6 votes
2 answers
848 views
The variety induced by an extension of a field
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 $k$...
9 votes
1 answer
357 views
Concerning $k \subset L \subset k(x,y)$
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 ...
- 2,875
2 votes
2 answers
258 views
General linear group action on extensions of finite fields
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 $n$-...
- 550
3 votes
0 answers
171 views
Bounds on degrees of minimal polynomials of infinite degree algebraic extension
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, $\deg m_\alpha$ ...
- 131
11 votes
3 answers
1k views
Cubic polynomials over finite fields whose roots are quadratic residues or non-residues
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 ...
user150240
6 votes
2 answers
565 views
Complete reducibility and field extension
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 ...
- 71
3 votes
0 answers
135 views
A bound for $[\mathbb{C}(x,y,z):\mathbb{C}(p,q,r)]$, where $\operatorname{Jac}(p,q,r) \in \mathbb{C}^{\times}$
Y. Zhang (in his PhD thesis) and P. I. Katsylo proved the following nice result; the two proofs are different, see: Zhang's thesis and Katsylo's paper: Let $f: (x,y) \mapsto (p,q)$ be a $k$-algebra ...
- 2,875
55 votes
0 answers
2k views
How many algebraic closures can a field have?
Assuming the axiom of choice given a field $F$, there is an algebraic extension $\overline F$ of $F$ which is algebraically closed. Moreover, if $K$ is a different algebraic extension of $F$ which is ...
- 41.6k
14 votes
1 answer
562 views
Sum of subfields of $\mathbb{C}$
Do there exist algebraically closed subfields $F_1, F_2, \dots, F_n$ ($n \geq 2$) of the field of complex numbers such that no $F_i$ is contained in $\bigcup_{j \neq i} F_j$ and $F_1 + F_2 + \dots F_n ...
- 151