3
votes
1answer
37 views

Field of Definition of an Algebraic Group

Linear Algebraic Groups- James E. Humphreys Chapter-XII Let $K$ be an algebraically closed field and $k$ be a arbitrary sub-field of $K.$ A closed set X in $A^n=K\times ...$(n times)$\times K$ is ...
2
votes
1answer
24 views

Sources about transcendence degree

I asked this question: Characterization of the transcendentals over a field I realized I need some knowledge about transcendence degree to prove some facts in the book I'm reading. I would like to ...
2
votes
1answer
34 views

complex automorphisms acting on a projective variety

Consider a complex projective variety $X=\operatorname{Proj}\frac{\mathbb C[T_1,\ldots,T_n]}{(f_1,\ldots,f_n)}$ with $f_1,\ldots,f_n$ homogeneous polynomials. If $\sigma\in\operatorname{Aut}(\mathbb ...
2
votes
1answer
43 views

Embedding a finite extension of $F(X)$ into a pure transcendental extension

If $F$ is any algebraically closed field, and $L \supset F(X)$ is a finite extension of the purely transcendental extension of $F$ of transcendence degree $1$, then can $L$ necessarily be embedded ...
1
vote
1answer
51 views

A question of algebraic geometry applied to field theory

I’ve come across this question in a coding theory course, and it has stumped me. Any hints and/or suggestions would be appreciated. Let $F$ be a field (for our purposes, assumed to be finite of ...
1
vote
1answer
38 views

Algebraic closure of a subfield of the field of fraction of a variety

I think that there is the following claim in Basic Algebraic geometry of Shafarevich. Let $X\to Y$ be a dominant morphism of varieties over an alebraically closed field $k$ of $char=0$. Let $\varphi: ...
2
votes
1answer
62 views

Isomorphism of an extension field of a field of finite transcendence degree

The following is the proposition (1.4) of Mumford's book Algebraic Geometry If $\mathbb C$ has infinite transcendence degree over $k$, then every variety has a $k$-generic point. In the proof ...
0
votes
0answers
22 views

$k(u)=k(t)\Leftrightarrow ht(u)=1$

In my lecture we proved the following statement: $u\in k(t)\backslash k$ show $k(u)=k(t)\Leftrightarrow ht(u)=1$ But I don't understand the proof we did (I'll put little numbers over the parts I ...
0
votes
2answers
45 views

$k$ algebraically closed field $\Rightarrow$ $V(f)\subset k^2$ infinite

Let $k$ be an algebraically closed field, and $f\in k[X,Y]$ a non-constant polynomial. Show that $V(f)\subset k^2$ is infinite. We solved this exercise in my tutorial class, but I have some questions ...
5
votes
1answer
73 views

Normal curve after base change (p > 0)

Suppose that $C$ is a normal projective curve over some base field $k$, possibly of positive characteristic. I am wondering to what extent one can modify the base field $k$ while not braking the ...
1
vote
1answer
53 views

Conditions under which a variety to remains smooth after base change (if p > 0)

Let $k$ be an arbitrary field of positive characteristic and let $V$ be a smooth projective (irreducible) variety over $k$. Suppose that $K/k$ is a field extension such that $V_K:=V\times_{\text{Spec ...
4
votes
0answers
64 views

Algebraical Fujita cancellation

If there exists some birational equivalence between ruled surfaces $C\times\mathbb{P}^1$ and $C'\times\mathbb{P}^1$ over a sufficiently nice field, then one can cancel the projective line and conclude ...
16
votes
1answer
384 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?
6
votes
2answers
195 views

Spec of tensor product of fields

Suppose $K/k$ is a finite separable extension of degree $n$. How to show that there exists a finite separable extension $k'/k$ such that $\operatorname{Spec}(K \otimes_k k') $ consists of $n$ ...
6
votes
2answers
255 views

Extension of residue fields and algebraic independence

Let $A$ be a Noetherian integral domain, $B$ a ring extension of $A$ that is an integral domain, $P \in \operatorname{Spec} B, \, p = P \cap A$. Denote by $\kappa(p),\ \kappa(P)$ the residue fields of ...
7
votes
1answer
108 views

Extension degree of residue field.

Let $k$ be a field, and $A$ be a finitely generated $k$-algebra with $\text{dim}(A)\leq 1$. Then for any maximal ideal $\mathfrak{m}$ of $A$, does this inequality $[A/\mathfrak{m}:k]<\infty$ hold? ...
9
votes
1answer
289 views

what is a “dévissage” argument?

In Serre's "Local Fields" (Chapter 2, section 2, proposition 3) he proves something about field extensions which he breaks into parts: first he dealt with the separable case, and then with the ...
10
votes
0answers
217 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 ...
3
votes
0answers
87 views

(Revised) Does p(a)=p(b) => a=b?

This is a revised version of problem posted here named :Does $p(a) = p(b) \Rightarrow a=b \ $? Let $S=(P_1,P_2,…,P_n)$ be a set of polynomials with complex coefficients. I call S critical if it ...
4
votes
2answers
97 views

Where do I use the fact that $F$ is algebraically closed in this proof?

I have to do the following. Let $F$ be an algebraically closed field. $I\in F[X_1,...,X_n]$ an ideal. Denote by $S(I)$ the subset in $F^n$ consisting of all $n$-tuples $(a_1,...,a_n)\in F^n$ such that ...
4
votes
1answer
124 views

Help with a bilinear form

Let $a,b\in\mathbb{F}_{2^{m}}$ (a field of characteristic $2$, m is odd) I need to prove that ...
0
votes
1answer
148 views

What is a valuation associated to an ordering on a field?

If $(K,\leq)$ is a totally ordered field with $P\!=\!\{\alpha\!\in\!K;\, 0\!\leq\!\alpha\}$, how is the valuation associated to $P$ defined? I was searching through Prestel & Delzell's Positive ...
2
votes
1answer
250 views

Two notions of uniformizer

Let $X$ be a projective algebraic curve and consider a `uniformizing' map $h:X \rightarrow \mathbb{P}^1$. Is there any connection between this notion of uniformizer and a uniformizer of the maximal ...
1
vote
0answers
94 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 ...
2
votes
2answers
252 views

Function field of variety

Let $k$ be a field, let $X/k$ be a geometrically integral variety and let $k(X)$ be its function field. If $L/k$ is an algebraic extension, is it true that the function field of $X \times_k L$ is $L ...
2
votes
1answer
221 views

algebraic closure and real closure are closure operators?

Are the algebraic closure (of a field) and the real closure (of a totally ordered field) closure operators (when restricted to appropriate sets of fields, so that they are maps on a set instead of a ...
10
votes
1answer
202 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 ...
10
votes
2answers
1k views

Luroth's Theorem

I have just begun to read Shafarevich's Basic Algebraic Geometry. In the first section of the first chapter, he quotes Luroth's theorem, which states that any subfield of $k(x)$ that is not just $k$ ...
0
votes
1answer
389 views

An attempt of catching the where-abouts of “ Mysterious group $Ш$ ”

This question is a bit concerned with the Tate-Shaferevich group, lets start defining $C$ as $$C: X^2- \Delta Y^2=4$$ which are generally called as Pell-conics, so all in this question $K$ refers to ...
16
votes
1answer
438 views

An exercise with Zariski topology

I read this exercise: Prove that the set $S = \{ (n, 2^n, 3^n ) \mid n \in \mathbb{N} \}$ is dense in $\mathbb{C}^3$ with Zariski topology. I have seriously thought about it, but I do not manage to ...
4
votes
1answer
152 views

On a characterization of the tamely ramified coverings of the fraction field of a strict Henselian ring

Let $K$ be the field of fractions of a strictly Henselian discrete valuation ring $A$. Following Milne ("Etale Cohomology", Chapter I, Paragraph 5, example e)), $Spec(K)$ is the algebraic analogue of ...