Questions tagged [field-theory]

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. Use (classical-mechanics), (electromagnetism), (general gravity) or (quantum-field-theory) for physical fields.

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47
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846 views

If polynomials are almost surjective over a field, is the field algebraically closed?

Let $K$ be a field. Say that polynomials are almost surjective over $K$ if for any nonconstant polynomial $f(x)\in K[x]$, the image of the map $f:K\to K$ contains all but finitely many points of $K$. ...
27
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0answers
743 views

Constructing an infinite chain of subfields of 'hyper' algebraic numbers?

This has now been cross posted to MO. 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 drawn ...
19
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1answer
624 views

Fields whose algebraic closure cannot be constructed without the axiom of choice

One can show that the statement that every field has an algebraic closure requires the axiom of choice. However, for almost all "everyday" fields, it seems that one can actually produce an algebraic ...
16
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0answers
380 views

If $K\cong K(X)$ then must $K$ be a field of rational functions in infinitely many variables?

If $k$ is any field, then the field $K=k(X_0,X_1,\dots)$ of rational functions in infinitely many variables satisfies $K(X)\cong K$ (by mapping $X$ to $X_0$ and $X_n$ to $X_{n+1}$). My question is, ...
16
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0answers
264 views

Find the cardinality of a subset of $GL_n( \mathbb F_p)$

Let $m,n \in \mathbb N$. Let $\mathbb F_p$ denote the prime field of characteristic $p$. Consider the set $$ X_m = \{A \in GL_n( {\mathbb F_p}): A^m=1 \}$$ Compute the cardinality of $X_m$. Its ...
16
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0answers
528 views

Relationship between intersection and compositum of fields

This issue came up in a number theory lecture today. Let $K$ be a number field and let $L/K$ be an abelian (finite Galois) extension. Then there exists a primitive $m$th root of unity $\zeta_m$ so ...
16
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0answers
364 views

Prop. 3.2.15 in Liu: Geometrically reduced algebraic variety

I have a problem with proposition 3.2.15 of Algebraic geometry and arithmetic curves of Qing Liu: Let $X$ be an integral algebraic variety over $k$. If $X$ is geometrically reduced then $K(X)$ is a ...
15
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0answers
538 views

Is there a field $F$ which is isomorphic to $F(X,Y)$ but not to $F(X)$?

Is there a field $F$ such that $F \cong F(X,Y)$ as fields, but $F \not \cong F(X)$ as fields? I know only an example of a field $F$ such that $F$ isomorphic to $F(x,y)$ : this is something like $F=k(...
15
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282 views

Does the category of algebraically closed fields of characteristic $p$ change when $p$ changes?

EDIT I've now posted this question on mathoverflow. It probably makes sense to post answers over there, unless someone prefers posting here. Let $\mathrm{ACF}_p$ denote the category of algebraically ...
14
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0answers
683 views

Limits and colimits in the category of fields

It is said that the category of fields $\mathsf{Fld}$ is ill-behaved, for example it is not an algebraic theory, does not have initial or terminal objects. In particular it is not presentable. On the ...
13
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237 views

The possible degrees of $\mathbb{Q}(a,b)$ in terms of the degrees of $a$ and $b$

Let $n,m \geq 1$ be natural numbers. Is there a characterization of those natural numbers $d$ for which there are algebraic numbers $a,b$ of degrees $n,m$ such that $\mathbb{Q}(a,b)$ has degree $d$ ...
13
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0answers
316 views

What is the degree of a real closure of an ordered field?

Given a subfield $F$ of $\Bbb R$, let the real-closure of $F$ be the smallest subfield of $\Bbb R$ which is real-closed and extends $F$. Since the theory of real-closed fields is complete, this is in ...
12
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0answers
149 views

What is the Galois group of the Hyperreal Numbers?

The Galois Group of $\mathbb{R}$ as an extension of $\mathbb{Q}$ is trivial. That’s because any field automorphism of $\mathbb{R}$ is order preserving, so since $\mathbb{Q}$ is dense in $\mathbb{R}$, ...
12
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0answers
93 views

Fields in which every odd-degree polynomial has a zero

An interesting fact about the field $\mathbb{R}$ is that it's halfway to being algebraically closed, inasmuch as every univariate odd degree polynomial with coefficients in $\mathbb{R}$ has a zero. ...
11
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488 views

algebraically closed fields of characteristic 0 and $\mathbb{C}$

Let $k$ be an algebraically closed field of characteristic 0. Then I've heard that if $k$ has cardinality no greater than that of $\mathbb{C}$, then there is an embedding $k\hookrightarrow\mathbb{C}$....
11
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0answers
91 views

Let $F_\infty=\bigcup_{n\geq1}\operatorname{Q}(2^{1/2^n})$ , $K_\infty=\bigcup_{n\geq1}\operatorname{Q}(\zeta_{2^n})$, what is the intersection?

Let $F_\infty=\bigcup_{n\geq1}\operatorname{Q}(2^{1/2^n})$ and $K_\infty=\bigcup_{n\geq1}\operatorname{Q}(\zeta_{2^n})$. What is the intersection $F_\infty\cap K_\infty$? (Here $\zeta_{2^n}$ is a ...
10
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0answers
106 views

Affine group, $[L : \mathbb{Q}] = n\varphi(n)$ or ${1\over2}n\varphi(n)$

Let $L$ be the Galois closure of $K = \mathbb{Q}(\sqrt[n]{a})$, where $a \in \mathbb{Q}$, $a > 0$ and suppose $[K : \mathbb{Q}] = n$. How do I see that $[L: \mathbb{Q}] = n\varphi(n)$ or ${1\over2}...
10
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642 views

On a Proof in Galois Theory

We have the following lemma (used in the proof of Abel's Theorem): If $\text{Char}(F)=0$, $E/F$ is a radical extension, and $K/E$ is the Galois closure of $E/F$, then $K/F$ is also a radical ...
10
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965 views

Deriving the Bring-Jerrard quintic using a $cubic$ Tschirnhausen transformation

One can simultaneously eliminate three terms from the general quintic using a quartic Tschirnhausen transformation. The 4th parameter of the quartic allows it to be done in radicals and details are in ...
9
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0answers
167 views

Every polynomial's image contains $0$ or $1$ in a field $\Bbb F$

This question talks about fields in which every polynomials are almost surjective, while I am interested in the following case: $\Bbb F$ is a field such that for every non-constant polynomial $f$ ...
9
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0answers
203 views

Does there exist a non-field Noetherian domain whose field of fraction is the field of real numbers?

Does there exist a Noetherian domain (but not a field) whose field of fractions is the field of real numbers $\mathbb{R}$ ? Any help will be appreciated. Thanks
8
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0answers
120 views

$m = [K(\alpha):K], n=[K(\beta):K]$ and $\gcd(m,n)=1$, prove that $K(\alpha + \beta) = K(\alpha,\beta)$

I am looking for an elementary demonstration of this: Suppose $K$ a field, $\mathrm{char}(K)=0$ and $K(\alpha) \supseteq K$ and $K(\beta) \supseteq K$ field extensions. Denote $n=[K(\alpha):K]$ and ...
8
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0answers
99 views

Sufficient conditions for $\mathbb{Q} (\alpha , \beta )=\mathbb{Q} (\alpha +c\beta ) \forall c\in \mathbb{Q}^*$

What are some sufficient conditions for $\mathbb{Q} (\alpha , \beta )=\mathbb{Q} (\alpha +c\beta ) \forall c\in \mathbb{Q}^*$ with $\alpha , \beta $ algebraic over $\mathbb{Q}$? We know that, for ...
8
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0answers
103 views

Field having exactly two extensions of each degree

It is well-known that a finite field has a unique extension of degree $n$, in a given algebraic closure, for every $n \geq 1$. Is there a field $F$ such that, in some given algebraic closure of $F$,...
8
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0answers
121 views

Integral domain over which every non-constant irreducible polynomial has degree 1

Let $R$ be an integral domain such that any polynomial $f(X) \in R[X]$ , which is irreducible in $R[X]$, has degree $1$. Then is it true that $R$ is a field ? If this is not true in general , What if ...
8
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0answers
291 views

An infinite prime can ramify right? (So what is Neukirch talking about?)

I have been under the impression for several years that if $L/K$ is an extension of number fields, then an infinite place of $K$ is said to ramify in $L$ if it comes from a real embedding of $K$ which ...
8
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0answers
164 views

Let $\mathbb{K} $ be a field of characteristic $p>0$ and $\mathbb{F} | \mathbb{K} $ a finite and separable extension.

Let $\mathbb{K}$ be a field of characteristic $p>0$ and $\mathbb{F}/ \mathbb{K}$ a finite and separable extension. Show that if $B=\{\alpha_1,\dots,\alpha_n\}$ is a basis, then $C=\{\alpha_1^p,\...
7
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0answers
132 views

What is an example of a non-simple finite extension $K/F$ such that the purely inseparable closure of $F$ in $K$ is simple?

The standard example of a finite extension that is not simple is to take $k$ to be a field of characteristic $p > 0$ and consider $k(x,y)$ over $k(x^p,y^p)$. In this case, the extension is purely ...
7
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0answers
122 views

What are the abelian extensions of $\Bbb C(X)$?

I would like to have a precise description of the finite abelian extensions of the field $ K = \Bbb C(X)$. Typically, can we describe the abelianization of its absolute Galois group $G_K$? Thoughts: ...
7
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0answers
121 views

Are there any other fields other than $\mathbb{R},\mathbb{C}$, rich enough to have analysis built on them?

I've been thinking about this, I don't know how to look up anything similar, so here I am asking a question. Specifically, is there any space $X$ with the following properties: Algebraic structure: ...
7
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0answers
114 views

Galois group of extension generated by all cubic roots

Let $K/\mathbb{Q}$ be generated by all cubic roots of rational numbers, that is $K=\mathbb{Q}(\{\sqrt[3]{a}:a\in\mathbb{Q}\})$. I would like to understand its Galois group. I only could prove that $K/...
7
votes
0answers
350 views

If $E/F$ is finite, $F$ has no odd extensions, and $E$ has no extensions of degree $2$, then $F$ is perfect and $E$ is algebraically closed?

I'm struggling to understand the claim that if $E/F$ is a finite field extension, $F$ has no odd extensions, and $E$ has no extensions of degree $2$, then $F$ is perfect and $E$ is algebraically ...
7
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0answers
104 views

Showing that poset of set of supports of a vector space is semimodular

Let $W$ be a subspace of the vector space $\mathbb{K}^n$, where $\mathbb{K}$ is a field of characteristic $0$. The support of a vector $v = (v_1,\ldots, v_n) \in \mathbb{K}^n$ is given by $\text{supp}(...
7
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0answers
584 views

Prove that every algebraic extensions over $\mathbb{R}$ are isomorphic to $\mathbb{R}/\mathbb{R}$ or $\mathbb{C}/\mathbb{R}$

Prove that every algebraic extensions over $\mathbb{R}$ are isomorphic to $\mathbb{R}/\mathbb{R}$ or $\mathbb{C}/\mathbb{R}$ Corollary $3.20$ page $267$ of Hungerford - Algebra: "Every proper ...
7
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0answers
161 views

When exactly is the splitting of a prime given by the factorization of a polynomial?

Let $L/K$ be an extension of number fields with $L=K(\alpha)$, where $\alpha\in \mathcal{O}_L$. Let $f\in \mathcal{O}_K[x]$ be the minimal polynomial of $\alpha$ over $K$. Let $\mathfrak{p}\...
7
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0answers
113 views

Ramification group fixing an unramified extension

For a Galois extension of local fields $L/K$ with Galois group $G$, define the ramification groups $G_i$ by $$G_i = \{ \sigma \in G : \nu_{L}(\sigma(x) - x) \geq i+1 \text{ } \forall x \in \mathcal{O}...
7
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0answers
540 views

Is there a general algorithm to find a primitive element of a given finite extension (with a finite number of intermediate fields)?)

I just want to know if there is an algorithm to find the primitive element of a given finite extension $F/k$ if the intermediate fields are given. I know how to approach it in particular examples ...
6
votes
0answers
115 views

An exercise about Galois theory

Let $L = \mathbb{C}(X, Y ,Z)$, $\omega$ a primitive 3rd root of unity, $\sigma: X \mapsto Y, Y \mapsto Z, Z \mapsto X$, $\tau: X \mapsto X, Y \mapsto \omega Y, Z \mapsto \omega^2 Z$, and $K$ the fixed ...
6
votes
0answers
35 views

Let $\mathbb F$ be a field, find a necessary and sufficient condition on $\mathbb F$ such that the only semi-polynomial maps are the polynomials.

Let $\mathbb F$ be a field. A map $f : \mathbb F^2 \rightarrow \mathbb F$ is semi-polynomial if for every fixed $y$ the map $ x \rightarrow f(x,y)$ is a polynomial and for every fixed $x$ the map $y ...
6
votes
0answers
47 views

First-order definability sums of squares

Let $K$ be a field. I am interested in when there can exist a first-order definition of the set $$ \Sigma K^2 := \lbrace \sum_{i=1}^n x_i^2 \mid n \in \mathbb{N}, x_1, \ldots, x_n \in K \rbrace $$ in $...
6
votes
0answers
97 views

Which $f,g \in \mathbb{C}[x,y]$ satisfy $\mathbb{C}(fy,gy)=\mathbb{C}(x,y)$?

Let $f=f(x,y), g=g(x,y) \in \mathbb{C}[x,y]$. Which $f,g$ satisfy $\mathbb{C}(fy,gy)=\mathbb{C}(x,y)$? Examples: (1) $f=2x$, $g=1-2x$. It is easy to see that $\mathbb{C}(2xy,(1-2x)y)=\mathbb{C}(...
6
votes
0answers
88 views

How to show that $G=\mathrm{Gal}[\Bbb Q(\xi_{2^n})/\Bbb Q(\xi_{2^m})]$ cyclic?

How to show that $G=\mathrm{Gal}[\Bbb Q(\xi_{2^n})/\Bbb Q(\xi_{2^m})]$ cyclic? $n>m>1$ are natural numbers. $\xi_{2^n}$ and $\xi_{2^m}$ are cyclotomic roots. I know that the order of $ G $ is ...
6
votes
0answers
106 views

Finding the splitting field of $f(x) \in \mathbb{Q}[x]$ in $\mathbb{C}$

Let $E$ be the splitting field for $x^4-2$ over $\mathbb{Q}$ in $\mathbb{C}$. I want to show that $E=\mathbb{Q}[i + \sqrt[4]{2}]$. I have a hint: Find at least five different elements in the orbit of ...
6
votes
0answers
1k views

Constructibility of the 17-gon

Comment: I greatly shortened and simplified the question. As a drawback, some comments/answers might not make any sense anymore. Assume we are using this set of axioms $A$ for plane euclidean ...
6
votes
1answer
600 views

When does a splitting field of a polynomial have the same degree as the polynomial?

Let's say we have the irreducible polynomial $f$ with roots $\alpha_1,\ldots,\alpha_n$. Now let $K$ be its splitting field, in other words $$K=\mathbb Q(\alpha_1,\ldots,\alpha_n).$$ When is it the ...
6
votes
0answers
156 views

Discrete valuation fields and representation as power series

Let $(K,v)$ be a discrete valuation field ($v$ is surjective). Let $\mathcal O$ be the ring of integers of $v$ and moreover let $\mathfrak p$ be the unique maximal ideal of $\mathcal O$. Then we have ...
6
votes
1answer
637 views

Showing that $\mathbb{Q}(\sqrt{2},\sqrt{3}, \sqrt{(9 - 5\sqrt{3})(2-\sqrt{2})})$ is normal over $\mathbb{Q}$, and finding its Galois group

If $K=\mathbb{Q}(\sqrt{2},\sqrt{3}, u)$, where $u^2 = (9 - 5\sqrt{3})(2-\sqrt{2})$, show that $K/\mathbb{Q}$ is normal, and find $\operatorname{Gal}(K/\mathbb{Q})$. I found that the minimal ...
6
votes
0answers
153 views

Is there an online database somewhere that lists identities for algebraic structures with two binary operators?

I'm working on an abstract algebra library in Python, and I'm trying to include as many functions that analyze algebraic structures, returning true or false based on whether or not the algebra ...
6
votes
0answers
314 views

Purely inseparable extension of algebraic function field

Let $K$ be a field with $\operatorname{char}(K) = p > 0$ and with the property that $[K:K^p] < \infty$. Let $F/K$ be an algebraic function field, i.e there is an element $x \in F$ which is ...
6
votes
0answers
148 views

Matrix representation of field automorphism

Let $K$ be the degree $n$ field extension of the field $k$, and let $\alpha_1,\dots,\alpha_n\in K$ be a basis of $K$ over $k$ (as a vector space). I read somewhere that the following matrix $$ M= \...