# Tagged Questions

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 ...
24 views

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 ...
34 views

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 ...
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 ...
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 ...
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 ...
53 views

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 ...
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 ...
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 $k$ ...
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 ...
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 ...
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 ...