# 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].

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### 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 $\mathrm{Gal}(\ell/k)$ ...
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### What's the number of irreducible polynomials over rational numbers with constraint on coefficients?

Suppose you have polynomials $f(x) = \sum\limits_{k=0}^n a_k x^k$ where $f \in \mathbb{Q}[x]$ and $a_k$ can be either $0$ or $1$. There are $2^n$ such polynomials in total. How can I find a number of ...
585 views

### Factorization of an irreducible polynomial in the field extension it defines

In field theory, the following fact is used in the construction of splitting fields: Given a field $F$ and an irreducible polynomial $f \in F[x]$, the quotient $F[\alpha]/(f(\alpha))$ is a field ...
145 views

### How to determine the degree of a rational function field over a relatively algebraic subfield?

Let $K$ be a field and $K(x_1,\cdots,x_n)$ be the degree-$n$ purely transcendental extension of $K$. Given homogeneous polynomials $f_1,\cdots,f_n\in K[x_1,\cdots,x_n]\setminus K$ with $\deg f_i=d_i$,...
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### Why is it so hard to give examples of differentially closed fields?

The theory of algebraically closed field, say in characteristic zero, and of differentially closed fields (of characteristic zero) have much in common: quantifier elimination and (hence) decidability; ...
112 views

### What direction does the derivation of an inseparable algebraic variable point in?

I've been thinking about the geometry of inseparable field extensions lately, since I'm studying smoothness in commutative rings in an advanced topics course this semester. I've generally come to the ...
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### In the Levi-Civita field, are there elements such that the standard parts of their subsequent powers produce an arbitrary sequence?

In the Levi-Civita field, are there elements such that the standard parts of their subsequent powers produce an arbitrary sequence? Particularly, is there an element $w$ of the field such that the ...
1 vote
77 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 ...
318 views

### Common Galois extension over $\mathbb Q$

Suppose $L'$ is a fixed cyclic galois extension over $\mathbb {Q}$ of degree $4$.Now we know that there exists also a degree $k$ extension $L$ over $L'$ but the extension $(L/\mathbb{Q})$ may not ...
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### A question about decompositions of rational functions

Let $f_1,g_1,f_2,g_2$ be non-constant rational functions on the Riemann sphere (i.e. elements of $\Bbb{C}(z)-\Bbb{C}$) satisfying $f_1\circ g_1=f_2\circ g_2$. Suppose there is a prime number $p$ such ...
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### Is the hierarchy of relative geometric constructibility by straightedge and compass a dense order?

Consider the hierarchy of relative geometric constructibility by straightedge and compass. Namely, given a geometric figure $B$, a set of points in the plane, we define that geometric figure $A$ is ...
980 views

### Algebraic topology over fields other than ${\bf R}$

Is there an algebraic topology for spaces defined on fields other than ${\bf R}$, including totally discontinuous fields ? By this, I am not talking about the field of coefficients, but about the ...