10
votes
3answers
189 views

Is there a better way to find the polynomial equation for this curve?

Consider the curve in $\mathbb{R}^2$ defined by the equation $$ x^{1/3} + y^{1/3} + (xy)^{1/3} = 1, $$ where $x^{1/3}$ denotes the real cube root of $x$, etc. Since the equation above involves only ...
3
votes
1answer
130 views

Action of a Galois Group on an Algebraic Variety

I've to solve the following exercise. Let's be $X$ a connected algebraic variety over $k$ and let $K$ be a finite Galois Extension of $k$ with Gaolis Group $G$. Now I have to prove that $G$ acts ...
16
votes
1answer
343 views

Is every rigid field perfect?

A field is rigid iff its automorphism group is trivial. A field $F$ is perfect iff all irreducibles in $F[x]$ are separable. Is every rigid field perfect?
4
votes
2answers
78 views

Is there a splitting field for multivariate polynomials over $\mathbb{Q}$?

Let $f(X,Y) = X^2 + Y^2 - c$. Does there exist a field $F$ such that $f(X,Y) = \prod_{i=1}^n(a_1^i X + a_2^i Y + a_3^i)$, where $a^i$'s are in $F$?
2
votes
0answers
76 views

Galois actions on extensions of algebraic function fields

Let $k$ be an algebraically closed field, $C/k$ and $C'/k$ be smooth projective curves, and $C'/k \rightarrow C/k$ be a $k$-morphism which is corresponding to the field extension $k(C) \hookrightarrow ...
3
votes
1answer
108 views

Galois group of $K(X)/K$

Let $K$ be an infinite field, if $K(X)$ is the field of rational function I want to find the Galois group of the extension $K(X)/K$. Lemma 1: If $L$ is a field such that $K\subsetneq L\subseteq ...
7
votes
1answer
281 views

Maximal ideals in polynomial rings over a field

Let $K$ be an algebraically closed field and let $k$ be a subfield of $K$ such that the field extension $K \mid k$ is algebraic. Let $B$ be the polynomial ring $K [x_1, \ldots, x_n]$ and let $A$ be ...
12
votes
2answers
138 views

Uniformly solvable families of polynomials

It is a famous theorem that there is no "quintic formula", i.e. there is no formula which expresses the roots of a quintic polynomial $x^5+a_4x^4+\cdots+a_0$ in terms of $a_4,\ldots,a_0$ and rational ...
6
votes
2answers
132 views

Usage of algebraic geometry in understanding the total Galois group of the rational

A professor of mine remarked in class that: "The tools via which we understand the total Galois group of the rationals comes from algebraic geometry". Could anyone shed some light on this remark, or ...
7
votes
1answer
198 views

Doesn't Abhyankar's lemma contradict faithful flat descent (no, but I'm confused)

Let $O_K$ be a dvr with fraction field $K$. Let $L/K$ be a tamely ramified finite Galois extension. Then, Abhyankar's Lemma implies that there exists a finite Galois extension $K^\prime/K$ such that ...
4
votes
0answers
81 views

Relationship between automorphisms of finite etale covers and function fields

Let $X$ and $Y$ be varieties over an algebraically closed field $K$, $\phi:Y \longrightarrow X$ be a finite etale cover , and let $K(X),K(Y)$ be the function fields of $X$ and $Y$ respectively. Then ...
1
vote
1answer
139 views

extension of algebraic function field

Let $K$ be a field, $t$ a transcendetal element over $K$ and $F'|K(t)$ an infinite Galois extension. Hence I have a tower of extension $K\subset K(t) \subset F' $. Does exist a subextension $F$ of ...
3
votes
1answer
97 views

etale fundamental group of a product

Let $X,Y$ be noetherian connected schemes over an algebraically closed field $k$ and let $\overline{x},\overline{y}$ be geometric points on them. There is a canonical homomorphism $\pi_1(X \times ...
2
votes
0answers
205 views

Serre's Topics in Galois Theory

I am supposed to write a running notes on Hilbert Irreducibility Theorem from Serre's Topics in Galois Theory as part of my master's Algebra course work. But I don't have any knowledge of Algebraic ...
10
votes
1answer
196 views

Finite extensions of rational functions

I know that finite extensions of $\mathbb{C}(x)$ correspond to finite branched covers of $\mathbb{P}^1$, and this leads to an abstract characterization of the absolute Galois group of $\mathbb{C}(x)$ ...
1
vote
0answers
89 views

Rational curve cover/Transcendental Galois field extension

Suppose the rational curve $C$ is a finite cover for the rational curve $D$ and the field of rational functions of $C$ is the purely transcendental extension $k(x)$ and that of $D$ is the subfield ...
7
votes
2answers
289 views

what is the simplest example of an etale cover which is not Galois?

By "Galois morphism" I mean a morphism $f: Y \to X$ such that $Y \times_X Y$ is a disjoint union of schemes isomorphic to $Y$. Let $X$ be reduced curve over afield of char. 0. I wonder what is a ...
5
votes
1answer
203 views

Why is Klein's quartic curve not hyperelliptic

Let $X$ be Klein's quartic curve given by $x^3y + y^3z+z^3x=0$ in $\mathbf{P}^2$. It is isomorphic to $X(7)$. How do I easily show that $X$ is not hyperelliptic? I can see that $X$ is of genus $3$ ...
3
votes
1answer
105 views

Are these two notions of Galois morphism the same

Let $f:X\to Y$ be a finite morphism of integral schemes. Let $G$ be the automorphism group of $X$ over $Y$. Are the following two conditions equivalent? The function field extension $K(Y)\subset ...
9
votes
1answer
194 views

Is $\operatorname{Gal}(\mathbb{Q}_p^{un})\cong \hat{\mathbb{Z}}$?

Is the absolute Galois group of $\mathbb{Q}_p^{un}$ the profinite completion of $\mathbb{Z}$? I was never quite sure... In similar cases, it is true. Namely, $\mathbb{C}((t))$ does have absolute ...
19
votes
2answers
1k views

Original works of great mathematician Évariste Galois

Through this question I wanted to know the original works of Galois. When I was reading Galois theory ( since from last month ) , I have been seeing one common line in every book, whose essence ...
7
votes
1answer
2k views

Reading the mind of Prof. John Coates (motive behind his statement)

To start with the issue, I have been thinking from many days that Birch-Swinnerton-dyer conjectures should have some association with the Galois theory, but one day I got the Article of Tate called as ...
2
votes
0answers
78 views

Is it true that $\forall G\leq Aut(\mathbb{P}^1)=PGL_2(\mathbb{C})$ the map $\mathbb{P}^1\rightarrow \mathbb{P}^1/G$ is defined over $\mathbb{Q}$?

Is it true that for every finite $G\leq Aut(\mathbb{P}^1_{\mathbb{C}})=PGL_2(\mathbb{C})$ the morphism $\mathbb{P}^1_{\mathbb{C}}\rightarrow \mathbb{P}^1_{\mathbb{C}}/G$ descends as a morphism (not ...