# Questions tagged [fields]

Fields as algebraic objects. For vector and tensor fields, use eg. [dg.differential-geometry]. For physical fields, use eg. [mp.mathematical-physics] or [quantum-field-theory].

**6**

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314 views

### $\mathbf{Q}_p$ versus $\mathbf{C}$

Let $\sigma_p : \mathbf{Q}_p\to\mathbf{C}$ and $\sigma_{\ell} : \mathbf{Q}_{\ell}\to\mathbf{C}$ be two field homomorphisms, with $\ell\neq p$.
Can one describe the compositum field $K$ of $\mathbf{Q}...

**14**

votes

**1**answer

371 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 ...

**5**

votes

**0**answers

100 views

### Constructing an infinite chain of subsets of 'hyper' algebraic numbers?

This question is cross posted from MSE.
Let $F$ be a subset of $\mathbb{R}$ and let $S_F$ denote the set of values which satisfy some generalized polynomial whose exponents and coefficients are ...

**4**

votes

**0**answers

130 views

### Field K and integer n such that the etale fundamental group of (GL_n)_K is the profinite completion of GL_n(K)?

Prove or disprove:
Claim: there exists a field $K$ and an integer $n$ such that $\pi_1^{et}((GL_n)_K)$ is isomorphic to the profinite completion of the abstract group $GL_n(K)$?
Note that then $G_K \...

**3**

votes

**0**answers

215 views

### Is the intersection of two function fields over finite fields again a function field?

I am interested in the question above. I know that the answer is NO if the base field is for instance $\mathbb{Q}$ (the intersection of $\mathbb{Q}(x^2)$ and $\mathbb{Q}((x-1)^2)$ is $\mathbb{Q}$ ...

**-1**

votes

**1**answer

248 views

### Write the algebra closure of $F_p$ as union of finite fields [closed]

This question follows Field theory by Steven Roman, Chapter 9, Exercise 20.
Denote the algebraic closure of the finite field $F_q$ by $\Gamma(q)$, and let $a_n$ be any strictly increasing infinite ...

**3**

votes

**0**answers

74 views

### What is known about the Hopf map for quadratic field extensions?

This question is related to my previous post:
Is this generalization of the Hopf map for quadratic field extensions surjective?
I still would like to know more and, while that post got several votes,...

**3**

votes

**0**answers

96 views

### Modal Principles of Field Extensions

In 2007 (with more work done later), J. Hamkins and B. Löwe found that the ZFC provably valid principles of forcing are the assertions of S4.2. In the introduction, they mention field extension as a ...

**0**

votes

**0**answers

109 views

### Is it possible to generalize a result of Katsylo-Zhang concerning the two-dimensional JC?

Let $p=p(x,y), q=q(x,y) \in \mathbb{C}[x,y]$ be a Jacobian pair, namely, $Jac(p,q)=p_xq_y-p_yq_x \in \mathbb{C}^{\times}$. Denote the total degrees of $p$ and $q$ by $\deg(p)$ and $\deg(q)$, and ...

**1**

vote

**0**answers

62 views

### Characterisation of projective modules over tensor products of fields

Let $k$ be a field and $L_1$ and $L_2$ finite field extensions of $k$.
Let $A:=L_1 \otimes_k L_2$ as an algebra.
Question:
Given a finitely generated $A$-module $M$, do we have that $M$ is ...

**0**

votes

**1**answer

78 views

### How to show the set $\operatorname{Hom}_K(L,\bar{K})$ of all $K$-embeddings of $L$ is partitioned into $m$ equivalence classes of $d$ elements each? [closed]

Let $L|K$ be a finite separable extension. Denote the algebraic closure of $K$ by $\bar K$.
$\forall x\in L$, denote $d=[L:K(x)]$ and $m=[K(x):K]$.
How to show the set $\operatorname{Hom}_K(L,\bar{...

**2**

votes

**1**answer

103 views

### Transitivity of an invariant of finitely generated field extensions

For a finitely generated extension of fields $K/k$, let us define "$S_{K/k}$" to be the minimum of the degrees $[K:\ell]$ where $\ell/k$ ranges over the purely transcendental subextensions of $K$ with ...

**2**

votes

**1**answer

159 views

### Having a separable extension of degree $n$ implies having a Galois extension of degree $n$?

I would like an explanation for the fact stated in the title. To repeat:
Question: How does one prove that if a field has a separable extension of degree $n$, then it has a Galois extension of ...

**3**

votes

**0**answers

198 views

### Eudoxus real numbers

I recently remembered the eudoxus construction of the real numbers.
Does anyone know what how the rational numbers $\mathbb Q$ can be characterised inside this construction?
Clearification: The ...

**1**

vote

**1**answer

225 views

### Formal Laurent and Taylor series

Let us define the field of formal Laurent series over a field $k$ as $K=k((x_1))((x_2))...((x_n))$. The subring of formal Taylor series $R=k[[x_1,...x_n]]$ is embedded in this field. Let us call its ...

**1**

vote

**0**answers

62 views

### Determinant and restriction of scalar

Let $E/F$ be a finite separable extension of fields, and $V$ a finite dimensional vector space over $E$. Let $T\in\operatorname{End}_EV$ be a linear operator on $V$, and let $\det(T)$ be its ...

**8**

votes

**1**answer

220 views

### Are all real-closed subfields of $\overline{\mathbb{Q}}$ conjugate?

Let $\overline{\mathbb{Q}}$ be the algebraic closure of $\mathbb{Q}$. The absolute galois group $G_\mathbb{Q}$ of $\mathbb{Q}$ acts on the set of real-closed subfields of $\overline{\mathbb{Q}}$.
...

**4**

votes

**0**answers

73 views

### Cohen structure theorem with explicit equations

By Cohen structure theorem, a complete regular equicharacteristic Noetherian local ring is isomorphic to a power series. In particular, this should hold for finite extensions of power series $k[[t]][\...

**2**

votes

**1**answer

144 views

### Is any real closed extension of $\mathbb R$ characterized up to isomorphism by its ladder?

Let $R$ be a real closed field. Recall that the ladder of $R$ is the divisible, ordered abelian group obtained by quotienting $R$ by a certain equivalence relation.
Note that $R$ has trivial ladder ...

**2**

votes

**1**answer

137 views

### Normal closure and separable elements

Let $K\subset E\subset\bar{K}$ be field extensions, $\bar{K}$ an algebraic closure of $K$. Denote $E_s$ the field of separable elements of $E$ over $K$, denote $\tilde{E}\subset\bar{K}$ the normal ...

**4**

votes

**1**answer

367 views

### Constructibility of the regular 17-gon [closed]

There is a standard construction of a regular heptadecagon by H.W. Richmond (1893) (https://en.wikipedia.org/wiki/Heptadecagon ) (As anyone knows, it was Gauss who found out that it is possible to do ...

**4**

votes

**0**answers

207 views

### Is there a converse of Abhyankar-Moh-Suzuki theorem?

The following question is the same as this question; I ask it here, since I have not got any comments there (I really apologize if I should have waited some more time before asking it here; it is just ...

**6**

votes

**1**answer

276 views

### Is this generalization of the Hopf map for quadratic field extensions surjective?

Let $k$ be a field, and let $L$ be a quadratic extension of $k$. Denote by $\sigma$ the non-trivial element of $\operatorname{Gal}(L/k)$. Let $M_2(L)$ be the vector space over $L$ of two-by-two ...

**2**

votes

**1**answer

161 views

### Alternate descriptions of finite fields

The finite field of order $p^n$ is isomorphic to $(\mathbb Z/p \mathbb Z)[X]/(P)$, where $P$ is an irreducible polynomial in $(\mathbb Z/p \mathbb Z)[X]$ of degree $n$. This describes every finite ...

**3**

votes

**1**answer

107 views

### 'Smallest' subfield of the Surreals which is isomorphic to the Surreals as an ordered group

What is the smallest subfield $F\subset N_0$ such that $$(F,+,\times,\leq)\ncong(N_0,+,\times,\leq)$$ but $$(F,+,\leq)\cong(N_0,+,\leq)?$$ Since these are all going to be proper classes cardinality is ...

**5**

votes

**1**answer

134 views

### When is a linear combination of the elementary symmetric polynomials reducible?

Let $n\ge 2$ and consider the polynomial ring $\mathbb F [X_1,...,X_n]$, where $\mathbb F$ is a field. Let $e_j:=e_j(X_1,...,X_n)$ be the elementary symmetric polynomial of degree $j$ in $X_1,...,X_n$...

**2**

votes

**0**answers

169 views

### Is the Fermat quartic surface a generalized Zariski surface?

Consider the Fermat quartic surface $$F\!: x^4 + y^4 + z^4 + t^4 = 0$$ over an algebraically closed field $k$ of odd characterstics $p$. Shioda proved that for $p=3$ this surface is a generalized ...

**2**

votes

**0**answers

43 views

### Lie algebra of derivations for a transcendental field extension and intersection fields

Suppose that $L$ is a finite Galois extension of the field $K$.
If $L_1$ and $L_2$ are subfields of $L$ containing $K$ then $L_1\cap L_2=L^H$
where $H$ is the group generated by ${\rm Aut}_{L_1}(L)$ ...

**13**

votes

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423 views

### Why does $E\otimes_KE\cong EG$ imply that Galois theory works?

This is a part of statement in the book I do not fully appreciate. Suppose $E/K$ is Galois extension and $G$ the galois group of $E/K$. $E[G]$ is the group ring formed by finite group $G$.
"It is ...

**1**

vote

**0**answers

112 views

### Factorially closed, finitely generated $k$-sub-algebra of $k[X_1,X_2,X_3]$ , where $k$ is algebraically closed field of positive characteristic

Let $S$ be a sub-ring of a commutative ring with unity $R$. Then $S$ is called factorially closed in $R$ if $a,b \in R$ and $ab \in S\setminus \{0\} \implies a,b \in S$.
Let $k$ be an algebraically ...

**1**

vote

**0**answers

46 views

### Factorially closed, finitely generated $k$-sub-algebra $A$ of $k[X_1,…,X_n]$, where $n>3$, $k$ is algebraically closed of char $0$, $trdeg_k A=n-1$

Let $S$ be a sub-ring of a commuttaive ring with unity $R$. Then $S$ is called factorially closed in $R$ if $a,b \in R$ and $ab \in S \implies a,b \in S$.
My question is : Let $k$ be an algebraically ...

**4**

votes

**1**answer

94 views

### Given a non-field local domain $R$, finding a dominating Valuation ring whose residue field is algebraic/finite extension of the residue field of $R$

Let $(R, \mathfrak m)$ be a non-field local domain with fraction field $Q(R)$ . Let $k_{R}:=R/m$.
We know that there is a Valuation ring $(V,\mathfrak m_V)$ such that $R \subseteq V \subsetneq Q(R)$ ...

**2**

votes

**0**answers

100 views

### What conditions make this embedding become algebraically closed?

This question is just out of curiosity.
Let $\mathcal{L}$ be the first-order language of theory of rings. Let K and F are two fields such that $K \subseteq F$. If K is existentially closed in F, ...

**5**

votes

**1**answer

143 views

### Is there a field with finitely many abelian extensions, that is neither separably closed nor real closed?

If $K$ has only finitely many Galois extensions, then $K$ must be either separably closed or real closed. Are there any other fields whose abelianizations are finite extensions (i.e. whose absolute ...

**2**

votes

**2**answers

122 views

### Formulas for the structure constants of a field extension basis given by a primitive element

Let $L/K$ be a finite separable field extension and let $\theta$ be a primitive element for $L/K$ with minimal polynomial $\mu(t) \equiv \mu_{\theta/K}(t) = \sum_{k=0}^n c_k t^k$. I am trying to ...

**9**

votes

**3**answers

472 views

### Algebraically closed field of cardinality greater than $\mathfrak c$

It is a 20th century result that there exists only one algebraically closed field for a given characteristic $p$ and cardinality $\kappa>\aleph_0$, up to isomorphism. Is there a better way to ...

**7**

votes

**2**answers

225 views

### Formally real fields with unique non-Archimedean ordering

My question is rather simple. Do there exist a formally real field that admits a unique ordering (so sums of squares are the positive elements) and such that this ordering is not archimedean?
Oh, I ...

**3**

votes

**0**answers

72 views

### Luroth's theorem for Discrete valuation rings?

Luroth's theorem states that if $k$ is a field and $L$ is a field extension of $k$ such that $k \subset L \subseteq k(X)$, then $L=k(f(X))$ for some $f(X) \in k(X) $ . My question is ; is there any ...

**19**

votes

**1**answer

679 views

### Is a field that never embeds twice in another field necessarily a prime field?

Call a field $k$ unrepeatable$^1$ if for every field $L$ there are either zero or one field homomorphisms $k \to L$. Then the prime fields $\mathbb{Q}$ and $\mathbb{F}_p$ for $p$ prime are clearly ...

**1**

vote

**1**answer

238 views

### On difference identities and $[K:F]$

Let $(K,\sigma)$ be a difference field of characteristic $0$, i.e. equiped with field morphism $\sigma:K\rightarrow K$. Assume that $K$ satisfy a non-trivial univariate difference polynomial identity $...

**2**

votes

**0**answers

85 views

### Indecomposablity in purely inseparable extensions

Let $k$ be a field of characteristic $p$ (e.g. the separable closure of $\mathbb{F}_p(t)$) and consider the extension $k(x)/k$ where $x^p\in k$ but $x$ does not. Consider a (finitely generated) ...

**8**

votes

**2**answers

307 views

### When is a bilinear form equivalent to a trace form?

Associated to a finite, separable field extension $L/K$, there is a natural nondegenerate bilinear form, the trace form, defined by $$\langle x,y \rangle := \mathrm{Tr}_{L/K}(xy)$$
Now, given a ...

**3**

votes

**0**answers

96 views

### Interpretation or application of this analog of minimal polynomial

Recently I was thinking about images of number field elements under a polynomial with coefficients in a smaller field, and I came across the following construction. It did not have the properties I ...

**2**

votes

**0**answers

95 views

### Fibers of reciprocity maps and higher dimensional analogs

Part I.
Say $K$ is a number field, $v$ is a finite place of $K$, $K_v$ the $v$-adic completion of $K$.
We have the local Artin map for every finite $v$:
$$\rho_v : K_v^{\times}\to\text{Gal}(K_v^{\...

**6**

votes

**2**answers

292 views

### Categoricity of the complex field in the generic extensions

Let $V[G]$ be a generic extension of $V$ by adding a new Cohen real (or generally a generic extension which adds new reals and do not blow up the power of the continuum). Working in $V[G],$ we can ...

**3**

votes

**0**answers

210 views

### Discrete vs. finitely generated subgroups of the adèles

If $U\subseteq\mathbf{R}^n$ is an additive subgroup, discrete with respect to the induced topology, then $U$ is a finitely generated abelian group.
Question.
Given a discrete additive subgroup $U\...

**0**

votes

**1**answer

101 views

### Galois Transcendental Field Extension has characteristic Zero

Let consider a Galois transcendental field extension $T/K$, therefore for each subextension $L$ of $T/K$ we have $T^{\operatorname{Aut}(T/L)} = L$.
My question is how to prove that this conditions ...

**2**

votes

**1**answer

158 views

### “Essential discriminants” of fields

Let $L$ be a number field, and let $K_1, \cdots, K_r$ be the maximal subfields of $L$ (that is, $K_j \subset L$ but for each $j$ there does not exist a proper subfield field $M_j$ of $L$ such that $...

**2**

votes

**0**answers

230 views

### Generalizations of Lüroth theorem

Let $k$ be an arbitrary field (I do not mind to take $k=\mathbb{C}$, if things are easier in this case).
A more general version of Lüroth theorem says that a field $L$, $k \subset L \subset k(x,y)$, ...

**2**

votes

**0**answers

155 views

### residue fields of smooth $\mathbf{Q}$-algebras

Let $A$ be a $\mathbf{Q}$-algebra. We say $A$ is "residually abelian", if there exists a maximal ideal $\mathfrak{m}$ of $A$ whose residue field $\kappa(\mathfrak{m})$ is a Kummer extension of an ...