# 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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### 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 $K$. ...
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### 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 ...
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### 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 ...
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### 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, ...
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### 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 ...
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 ...
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 ...
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### 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 ...
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 ...
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### 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 $f$ ...
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
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### $m = [K(\alpha):K], n=[K(\beta):K]$ 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 ...
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### 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 ...
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 $F$,...
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 ...
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 ...
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### 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 ...