# Tagged Questions

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I am a senior undergraduate student in mathematics, I have a sound knowledge in the following areas: a) Commutative Algebra b) Field Theory and Galois Theory c) Homological Algebra My question is ...
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### roots of cubic polynomial

On page 26 of Milne's Elliptic Curves (http://www.jmilne.org/math/Books/ectext5.pdf), he states the following: "... a cubic polynomial $h(x) \in k[x]$ with two roots in $k$ has all of its roots in ...
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### Does the data of Galois group, ramified places, and inertia groups, determine a Galois number field?

Suppose I tell you that $K/\mathbb{Q}$ is a finite Galois extension, and I specify the Galois group $G$, and suppose further that I give you a finite list $S$ of places of $\mathbb{Q}$ and for each ...
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### Locally finite infinite field

Is there a place (a book maybe) where I can find some useful information on infinite locally finite fields? Especially when all of whose proper subfields are finite? I know, for instance, that a ...
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### What does $K^{1/p}$ for a field $K$ mean?

In the proof of the finite generation of the invariant ring of a finite group acting on $k[x_1,\dots,x_n]$, at one time there is a symbol I don't understand. The situation is as follows. $k$ is a ...
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### Counterexamples in algebra

I got the feeling that whenever a subject gets so sophisticated that Zorn's lemma is needed, a book of counterexamples in that subject would probably benefit researchers/ students a lot. Zorn's ...
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### Abstract algebra book recommendations for beginners.

I am taking abstract algebra in this semester. I find that my lecturer didn't provide enough examples for understanding. So I would like to ask what are the recommended books which has lots of solved ...
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### When are powers of primitive elements still primitive elements

This question is motivated by this question and is tangentially related to this question. Let $L/K$ be a finite Galois extension of fields. Pick $\alpha \in L \setminus K$ and consider the simple ...
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### Generators for $\mathbb F_p^*$

Let $p$ be prime, then it is a well-known fact that $\mathbb F_p^*= \mathbb F_p -\{0\}$ is a cyclic group under multiplication. Are there any methods to determine the generators of this cyclic or any ...
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### n-th roots of unity form a cyclic group in a field of characteristic p if gcd(n,p) = 1

Let $n$ be a positive integer, and let $\mathbb F$ be a field of positive characteristic $p$ with $\gcd(n,p) = 1$. Where can I find some proofs that the group of all $n$-th roots of unity (in an ...
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### Cyclic Algebras over local fields- reference request

I am looking for an introductory text to the subject of Cyclic Algebras, and in particular ones defined over a local field. A cyclic Algebra, to the best of by understanding is defined as follows: ...
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### When, and by whom, was “$\mathbb{C}$ is algebraically closed” dubbed the “fundamental theorem of algebra”?

Wikipedia has this enigmatic sentence on the page for the fundamental theorem of algebra: ...its name was given at a time when the study of algebra was mainly concerned with the solutions of ...
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### 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 ...
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### Characterization of fields $\mathbb{F}$ with the property: there exists a vector space $V$ such that $V^{\star}$ is isomorphic to $\mathbb{F}[X]$

In this post, Georges Elencwajg says that Erdős-Kaplansky tell us that there does not exist a real vector space whose dual is isomorphic to R[X] ! I would like to know if there is a ...
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### A field which is not algebraically closed but has no extensions of a fixed degree(s)?

Consider the field $k$ obtained as the union of all finite towers of degree $2$ extensions over the rationals. Then $k$ has no degree $2$ extensions, yet $k$ admits extensions of every other finite ...
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 ...