8
votes
0answers
83 views
+50
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
75 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
91 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
84 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
...
1
vote
1answer
102 views
Transcendence degree for algebra $R$ over $k$
Let $R$ be an integral domain over a field $k$.
Is it true, that $deg.tr_k \, Frac(R)$ is the greatest
number of elements of $R$ algebraically independent over $k$?
0
votes
1answer
139 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
124 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
69 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
133 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
149 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 ...
8
votes
1answer
144 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
626 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
360 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 ...
14
votes
1answer
360 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 ...
3
votes
1answer
130 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 ...
