# Tagged Questions

79 views

### Any abstract algebra book with programming (homework) assignment?

All: I had studied abstract algebra long time ago. Now, I would like to review some material, particularly about Galois theory (and its application). Can anyone recommend an abstract algebra book ...
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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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### Learning about Grothendieck's Galois Theory.

I have background in category theory and I am familiar with the very basics of algebraic geometry - Chapters I and II of Hartshorne. What would be a recommended (self-contained, maybe?) text for ...
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### Question about inverse Galois problem

I have a question... if for every finite simple group, we can construct a Galois extension over $\mathbb{Q}$ with that Galois group, does it follow that we can construct Galois extensions over ...
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### Help with Polynomial Roots Problem

Let's consider the case of two variables, $p\in\mathbb{R}[x,y]$. Suppose I want to find when there is $c\in\mathbb{R}$ such that p(x,x)+p(x,c-x)-p(c-x,x)-p(c-x,c-x)=0 \textbf{ for all } ...
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### Proof of Fermats Last Theorem for Given Exponent

Where can I find reasonably short and elementary proofs (using basic concepts of ring, field, galois theory) of Fermat's Last Theorem for specific n? For example, $n=5,7,13$?
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### Best book ever on Galois theory (and differential galois theory) [closed]

Which is the single best book for Galois theory (that includes differential Galois theory) that everyone who loves pure Mathematics should read?
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### Semialgebraic conditions that convey properties of Galois group

Let $f \in \mathbb{Z}[x]$ be a polynomial of degree $n$ with integer coefficients and let $G_f$ be the Galois group of $f$ over $\mathbb{Q}$. I am trying to collect results that convey some ...
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I have always been interested in the Inverse Galois Problem since I read the story of Évariste Galois. I have recently finished up courses in rings, fields, groups, and finally Galois theory. I really ...
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### Classes of groups known to be realizable (IGP)

A finite group $G$ of order $n$ is said to be realizable (over $\mathbb{Q}$) if there exists a Galois extension $L/\mathbb{Q}$ such that $\mathrm{Gal}(L/\mathbb{Q})=G$. I'm curious what classes of ...
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### 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 ...
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### presentation of the inertia group of $p$-adic fields

It is known (see here) that the absolute Galois group of a $p$-adic number field $K$ (ie a finite extension of $\mathbb Q_p$) is topologically finitely presented for any odd prime $p$. Is it also ...
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### Self teaching Galois Theory

At uni, I did a module in group theory which I really enjoyed. I also did one on number theory which had aspects of rings and fields in it and I enjoyed learning this. Now that I have finished uni, I ...
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### Maximal ideals in polynomial rings over a field

Let $K$ be an algebraically closed field and let $k$ be a subfield of $K$ such that the field extension $K \mid k$ is algebraic. Let $B$ be the polynomial ring $K [x_1, \ldots, x_n]$ and let $A$ be ...
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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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### On Galois Theorem and Dummit and Foote text !

in dummit and foote text , galois theory is presented in chapter 14 . group theory is presented in 6 chapter , ring theory in 3 chapter and so on my qestion is , which chapters of the text is ...
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### Necessarily complex analytic proofs in algebra.

Does anyone know of an example where complex analysis is necessary to prove something in algebra? I would be particularly interested in results from group theory or Galois theory. In an ideal ...
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### Inverse Galois problem for small groups

I am looking for a list of all small groups (maybe order $\leq 20$) realized as the Galois groups of a polynomial over $\mathbb{Q}$, with proof. Any idea where I could find these? Partial answers or ...
Choose a random polynomial $P\in\mathbb{Z}[x]$ of degree $n$ and coefficients $\leq n$ and $\geq-n$. Let $r_1,\ldots,r_n$ be the roots of $P$ and consider ...