1
vote
2answers
47 views

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 ...
5
votes
1answer
35 views

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 ...
1
vote
1answer
62 views

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 ...
1
vote
1answer
37 views

What are the roots of unity quadratic integers?

The article of Roots of unity in wikipedia implies that the following roots of unity are quadratic numbers: $$ \{\pm 1\}, \{\pm 1,\pm i\}, \{\pm 1,\pm \zeta,\pm \zeta^2\}. $$ where $\zeta=\exp(2\pi ...
4
votes
2answers
81 views

Primitive elements for $K=\Bbb{Q}(\sqrt{2},\sqrt{3})$

The key lemma for proving the primitive Element Theorem (for finite extension of a field $F$ with characteristic $0$) in Artin's Algebra (2nd edition) is the following: Suppose $char F=0$ and ...
0
votes
1answer
44 views

book on cubic fields by Kisilevsky

I have been trying to get a copy of Indices in cyclic cubic fields, in “Number Theory and Algebra”, Academic Press, 1977 by D. S. Dummit and H. Kisilevsky but it is nowhere to be found. Does anyone ...
5
votes
1answer
91 views

Ramification in the ring of all algebraic integers

If $F$ is a finite extension of $\mathbb{Q}$ then its of integers $R$ is a Dedekind domain, and has unique factorization of ideals into powers of prime ideals. For each prime number $\ell$, you can ...
3
votes
2answers
444 views

source to learn Galois Theory

What kind of recommendations do you have for a very good source to learn Galois Theory? Is there any Atiyah-MacDonald-type book on Galois theory? What is your opinion on the chapters from Lang and ...
0
votes
2answers
52 views

Field homomorphism respects arbitrary arithmetic expression

A field homomorphism $f:A \to B$ respects the fields' binary operations, in the sense that $f(x + y) = f(x) + f(y)$ and $f(xy) = f(x)f(y)$. When you have an explicit expression like $expr = x^3 + 15x ...
2
votes
1answer
55 views

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 ...
6
votes
2answers
340 views

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 ...
5
votes
6answers
1k views

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 ...
3
votes
1answer
78 views

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 ...
3
votes
1answer
71 views

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 ...
1
vote
4answers
630 views

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 ...
2
votes
1answer
142 views

Automorphism extension property of Galois extensions

If $L$ is a Galois extension of $K$ and $M$ is a finite Galois subextension of $L \mid K$, then a standard lemma says that any automorphism of $M \mid K$ can be extended to an automorphism of $L \mid ...
0
votes
1answer
54 views

What is known about moduls $M = F^n$ over a ring $R$ where $F = R/I$ is a field

If $R$ is a ring and $I$ is an ideal of $R$, then $F = R/I$ is a homomorphic image of $R$, i.e. there is a homomorphism $f: R \rightarrow F$. If you let $M = F^n$, and define $(\cdot): (R,M) ...
3
votes
1answer
177 views

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: ...
19
votes
1answer
357 views

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 ...
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 ...
1
vote
1answer
49 views

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 ...
8
votes
1answer
93 views

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 ...
8
votes
2answers
261 views

Good undergraduate level book on Cyclotomic fields

I have Lang's 2 volume set on "Cyclotomic fields", and Washington's "Introduction to Cyclotomic Fields", but I feel I need something more elementary. Maybe I need to read some more on algebraic number ...
3
votes
1answer
150 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 ...