The abstract-algebra tag has no wiki summary.
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Solving a system of rational functions
Given pairwise distinct numbers $c_1, c_2, \dots c_n \in \mathbb{C} \setminus \{0\}$, does the system of equations $$\frac{6}{c_k} + \sum_{i \ne k} \frac{2}{c_k - c_i} = \sum_{i = 1}^n \frac{1}{c_k - ...
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47 views
What is the amount of abstract algebra needed to study elliptic curves? [migrated]
To be more specific, how much is needed to understand the book 'Rational points on elliptic curves' by Silverman?
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0answers
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Commutativity of ring of order $p^2$ with unity $e$ and characteristic $p$ [migrated]
Let $R$ be a finite ring of order $p^2$ with unity $e$ and characteristic $p$. This ring is commutative but I cannot get why it is.
I know that this ring looks as $\mathbb Z /p\mathbb Z \times Z ...
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2answers
349 views
Shape of axioms in abstract algebra
When defining abstract algebraic structures (like monoids, groups, etc...), are there some constraints on the shape of the axioms, for the structure to have good properties that we implicitly use in ...
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205 views
$\mathbb{Z}G$ (left) Noetherian$\Rightarrow$ $l^1(G)$ is a flat $\mathbb{Z}G$-(right) module?
Let $G$ be a countable discrete group (not necessarily abelian), and suppose the group ring $\mathbb{Z}G$ is a left-Noetherian ring, for example, when $G$ is a polycyclic-by finite group.
Denote the ...
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0answers
80 views
Lazard's $\Gamma_n(f)$ as cocycle
In Michel Lazard's "Commutative Formal Groups" Springer Lecture Notes, he defines an operator on a polynomial 3-cochain $f$ denoted $\Gamma_n(f)$, which defines as the $n^{th}$ homogeneous piece of ...
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Show that a nonabelian group of ordr p^n has an automorphism of order p. [migrated]
I am not sure how to prove this question: "Show that a nonabelian group of order p^n has an automorphism of order p". I found a proof of Wolfgang Gaschütz that if G is a finite non-abelian p-group, ...
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1answer
190 views
Why do no prime ideals ramify in the extension $\mathbb{Q}(\sqrt{p }, \sqrt{q})/\mathbb{Q}(\sqrt{pq })$? [closed]
Let $p,q $ be odd integer primes, $p \equiv 1 \pmod 4$ and $q \equiv 3 \pmod 4$.
$K = \mathbb{Q }[\sqrt{pq }]$, $L = \mathbb{Q}[\sqrt{p }, \sqrt{q} ]$. Why a prime ideal in $O_{K}$ doesn't ramify in ...
1
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1answer
75 views
“Symmetric” Polynomial 4-cocycles
It is an old theorem of Heaton's (based on work of Eilenberg and MacLane), that a polynomial 3-cocycle $f(x,y,z)$ which is "symmetric," in the sense that $f(x,y,z)-f(x,z,y)+f(z,x,y)=0$, is always a ...
2
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1answer
115 views
English translation of Steinitz 1910?
Does there exist an English translation of Steinitz' 1910 work "Algebraische Theorie der Körper"?
http://www.digizeitschriften.de/dms/img/?PPN=GDZPPN002167042
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90 views
classification of rank $2$ $\mathbb{Z}/p^n\mathbb{Z}$-algebra with invertible discriminant
Let $p$ be a prime number and $n$ be an integer. Let $A$ be an $\mathbb{Z}/p^n\mathbb{Z}$-algebra of rank $2$ whose discriminant is non invertible. In Serre's book lecture on the mordell Weil theorem ...
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1answer
129 views
Find a special element in group algebra
Let $$G=\langle x, y, z\mid xyx^{-1}=zy, xzx^{-1}=z, yz=zy\rangle,$$ denote $l^1(G)^{\times}$ to be the set of units in $l^1(G)$, which we have considered as a ring with multiplication defined by the ...
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48 views
Example of a ring whose minimals are annihilators of idempotents?
I'm looking for examples† of rings with the property that for each
$P={\rm Ann}_R(a)\in{\rm Min}(R)$ then $a\in R$ is idempotent (ie $a^2=a$)
† other than domains!
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0answers
102 views
Is a certain group related to a primitive L function isomorphic to $Gal(\overline{\mathbb{Q}}_{\ell}/\mathbb{Q}_{\ell})$ for some $\ell$?
I define the notion of "Galois class of L functions" in the following way:
$A$ is a Galois class of L functions if and only if the follwing three conditions hold simultaneously:
1) every element ...
3
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1answer
176 views
Diagonalize the simultaneous matrices and its background
For two $n \times n$ nonnegative definite Hermitian matrices $A$ and $B$ over the real number field $\mathbb R$:
Question1:Is there always a
nonsingular matrix $P$ over the same
field $F$ ...
