Abstract algebra is the study of algebraic objects. Some of the more common algebraic objects are groups, rings, fields, vector spaces, modules, and other advanced topics.

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391 views

Why is such an ideal ambiguous?

Suppose I have an $R$-ideal $I$ with $$I=(1-\zeta)^n XR$$ with $R=\mathbb{Z}[\zeta]$, $\zeta$ a primitive $p$-th root, $X$ an ideal in the integral closure of $\mathbb{Z}$ in $\mathbb{Q}[\zeta + ...
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vote
1answer
101 views

$k$-basis of a quotient of ideals in polynomial ring

Let $k$ be a field and consider the ideal $I=(x,y) \subset k[x,y]$. Am I correct in saying that $(x,y)/(x,y)^{2}$ is generated as a $k$-vector space by the class of $x$ and $y$?
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1answer
176 views

The divided polynomial algebra over a field

Let $\Gamma_R[\alpha]$ denote the divided polynomial algebra over $R$; that is, the quotient of the free $R$-algebra $R\langle \alpha_1,\alpha_2,\cdots \rangle$ by the relations $$\alpha_n \cdot ...
1
vote
1answer
214 views

Finite subgroup

In the following problem, the only parts I didn't understand were c) and e). The remaining I did. Please, help me! Let $A,B$ be subgroups of $G$ that normalize each other. Assume that the set ...
2
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3answers
305 views

Relating $\operatorname{lcm}$ and $\gcd$

I would appreciate help to show this equality is valid: $\operatorname{lcm} (u, v) = \gcd (u^{- 1}, v^{- 1})^{- 1}$, where $u, v$ are elements of a field of fractions. In the text it is stated that ...
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1answer
271 views

Proper way to create Goppa code check matrix?

I'm trying to figure out what is the correct way to compute a check matrix for a binary Goppa code. So far (searching through the publications) I've found more than one possibility to do that, and I'm ...
1
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1answer
103 views

Intuition surrounding units in $R[x]$

My lecture notes state that an 'easy' result is If $R$ is an integral domain then an irreducible element of $R$ remains irreducible in $R[x]$, and the units in $R$ and in $R[x]$ are the same. I ...
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1answer
67 views

Proving that when $\nu(x) = \nu(y)$ in the Gaussian integers they are associates

I think I'm missing something here but my lecture notes just seem to state that 'clearly' for $x$ and $y$ in the Gaussian integers (elements of $\mathbb{Z}[i]$, a Euclidean domain), if $\nu(x) = ...
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2answers
128 views

The definition of a unique factorisation domain

Wikipedia gives the definition of a Unique Factorisation Domain as one where every element "can be written as a product of prime elements (or irreducible elements)" which suggests that in a UFD prime ...
2
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1answer
753 views

Subgroups of finite solvable groups. Solvable?

I am attempting to prove that, given a non-trivial normal subgroup $N$ of a finite group $G$, we have that $G$ is solvable iff both $N$, $G/N$ are solvable. I was able to show that if $N,G/N$ are ...
3
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0answers
224 views

Double Coverings of the Double Torus

I'm trying to count all the double coverings of the double torus. I know that the fundamental group of the double torus is $$\pi_1(X)=\langle a,b,c,d;[a,b][c,d]\rangle $$ where ...
2
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1answer
203 views

Generators for $PSL(2,\mathbb{Z})$ with a specific property

Does there exist generators $S$ and $T$ for the modular group $\Gamma=PSL(2,\mathbb{Z})$ with the following property: $$S+S^{-1}+T+T^{-1}=0$$ Here is a candidate: $$S=\left[\begin{array}{cc} -1 ...
6
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4answers
372 views

What exactly is an $R$-module?

From Wikipedia: "If $R$ is commutative, then left $R$-modules are the same as right $R$-modules and are simply called $R$-modules." The definition of left $R$-module: $M$ is a left $R$-module if $M$ ...
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2answers
844 views

What exactly is a tensor product?

This is a beginner's question on what exactly is a tensor product, in laymen's term, for a beginner who has just learned basic group theory and basic ring theory. I do understand from wikipedia that ...
3
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1answer
176 views

Proof that a polynomial is irreducible in $\mathbb{Q}$

Let $p$ be a prime number, and let $m,k_1,\ldots,k_{p-2}$ be even numbers. Define the polynomial $h(x)=(x^2+m)(x-k_1)\cdots(x-k_{p-2})$ and $r=\min \{|h(a)|\mid a\in\mathbb{R},h'(a)=0\}$. Under these ...
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1answer
113 views

In general rings, what do principal ideals look like?

For some ring $R$ (no $1$ or commutativity necessary) and $a \in R$, we defined the principal ideal $(a)$ by $$ (a) := \bigcap \{ I : a \in I \subseteq R, I \text{ is an ideal}\}.$$ Now as a ...
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1answer
314 views

Why is this ideal projective but not free?

Let $R=\mathbb{Z}[\sqrt{-5}]$ and $I=(2,1+\sqrt{-5})$. How can I prove that $I$ is projective but not free?
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1answer
204 views

Isomorphic group to the multiplicative group of a field.

Suppose that I have an abelian group $G$ and an epimorphism from $G$ to $(K^*)\times (K^*), $ where $K$ is an algebraically closed field. Is it true that $G$ cannot be isomorpic to $K^*.$?
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5answers
276 views

Write down a $\mathbb Q$-Basis for $\mathbb Q(\alpha, i)$ and show that $\mathbb Q(\alpha, i) = \mathbb Q(\alpha + i)$.

I have found a $\mathbb Q$-basis for $\mathbb Q(\alpha)$, where $\alpha$ is a root of $x^{3}-x+1$, to be $\{1, \alpha, \alpha^{2}\}$ & a $\mathbb Q(\alpha)$-basis for $\mathbb Q(\alpha, i)$ to be ...
0
votes
1answer
229 views

Cyclic Group Questions

I have two questions, the type of which I always struggle to answer. These are questions from some old papers. I've tried to find out from my textbook how to handle these $x^a = e$ questions, but I ...
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1answer
185 views

Question about a property certain algebraic extensions $E/K$ (not necessarily separable) have.

A few days ago I found this question here on math.stackexchange, which gave a sufficient criterion for a separable, algebraic extension $E/K$ to be an algebraic closure of $K$. However it was claimed ...
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3answers
848 views

Galois Group of $(x^3-5)(x^2-3)$

I am having some trouble calculating the Galois group (over $\mathbb{Q}$) of $(x^3-5)(x^2-3)$. I can see the splitting field is ...
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2answers
592 views

Show the norm map is surjective

Let $K/F$ be an extension of finite field. Show that the norm map $N_{K/F}$ is surjective. Here is what I have so far: Since $F$ is a finite field and $K/F$ is a finite extension of degree $n$, so ...
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2answers
85 views

Integrally closed with roots of identity

Let $\lambda_1,...,\lambda_n$ be roots of unity with $n\geq 2$. Assume that $$\frac{1}{n}\sum_{1}^{n}\lambda_i$$ is integral over $\mathbb{Z}$. Show either $\sum_{1}^{n}\lambda_i=0$ or ...
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2answers
309 views

Approximation Lemma in Serre's Local Fields

Let $A$ be a Dedekind domain, and let $K$ be its field of fractions. In Serre's Local Fields, the following Lemma is stated. Approximation Lemma Let $k$ be a positive integer. For every $i$, ...
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1answer
798 views

An abelian group of order 100

The first part of the problem asks you to prove that an abelian group $G$ with order $100$ must contain an element of order $10$. For this part, I use Sylow theorem to list possiblities for $H$ and ...
4
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1answer
156 views

Algorithms for symbolic definite integration?

What are the algorithms for symbolic definite integration? Apart from computing the antiderivative first. What are the basic ideas behind such algorithms? As far as I got it, the main idea behind ...
3
votes
3answers
97 views

$N$ submodule of $M$ and $N \cong M$ does not necessarily imply that $M=N$

Let $M, N$ be $A$-modules with $A$ being a commutative ring. Suppose that $N$ is a submodule of $M$ and also that $N$ is isomorphic to $M$. According to my understanding this does not necessarily ...
2
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1answer
75 views

$S^{-1}M \cong S^{-1}N$ does not imply $M \cong N$

Let $M, N$ be $A$-modules, where $A$ is a commutative ring with identity. Let $S$ be a multiplicative subset of $A$ that contains no zero divisors and contains the identity of $A$. I am looking for a ...
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1answer
84 views

The size of Galois group

In continuation to my last post: In class we saw an example that says: $n=[\mathbb{F}_{p^n}:\mathbb{F}_{p}]=|\operatorname{Gal}(\mathbb{F}_{p^n}/\mathbb{F}_{p})|$ ; ($p$ is prime). My thoughts are ...
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2answers
335 views

Is the size of the Galois group always $n$ factorial?

I am studying field theory, and I just started the chapter on Galois theory. Since a Galois extension is the splitting field of some polynomial $p(x)$ and this polynomial have exactly $n$ roots in ...
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2answers
121 views

Homogeneous polynomials

Let $f$ be a homogeneous polynomial of degree $k$ in $K[x,y]$ where $K$ is a field, let $g$ be a homogeneous polynomial of degree $j$ in $K[x,y]$ such that $f(x,y)/x^{k} = g(x,y)/y^{j}$. Why this ...
1
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1answer
216 views

Why is the cyclotomic polynomial over $\mathbb{Q}$?

As defined in Wikipedia (and this is the same definition I was given in class), it is not clear to me why the cyclotomic polynomial is over $\mathbb{Q}$. It is over $\mathbb{C}$, but I don't see a ...
3
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4answers
429 views

If $I = \langle 2\rangle$, why is $I[x]$ not a maximal ideal of $\mathbb Z[x]$, even though $I$ is a maximal ideal of $\mathbb Z$?

Let $I = \langle 2\rangle$. Prove $I[x]$ is not a maximal ideal of $\mathbb Z[x]$ even though $I$ is a maximal ideal of $\mathbb Z$. My professor mentioned that I should try adding something to ...
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2answers
330 views

Coproduct in the category of (noncommutative) associative algebras

For the case of commutative algebras, I know that the coproduct is given by the tensor product, but how is the situation in the general case? (for associative, but not necessarily commutative algebras ...
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2answers
232 views

For any group $\mathrm{ord}( a_1\circ a_2\circ\dots\circ a_{n-1}\circ a_n)=\mathrm{ord}(a_2\circ a_3\cdots\circ a_{n-1}\circ a_n\circ a_1)$

I would appreciate some advice as how to start with the following problem: Show through induction that in every Group G, for all $a_1,a_2,a_3,..., a_n$ that: $$\mathrm{ord}( a_1 \circ a_2 \circ ...
3
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1answer
411 views

Definition of Separability Degree

For an assignment, I am trying to determine the separability degree of some algebraic field extension $L/K$. The definition of the separability degree of polynomial is not difficult to find at all, ...
6
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1answer
216 views

Problem about the definition of Euclidean domain

In the definition of domain, we first define a degree function $\vartheta: R^\times \rightarrow \mathbb{N}$ with such two constraints: (1) $\vartheta(f)\leq \vartheta(fg)$ for all $f,g\in R^\times$. ...
3
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1answer
150 views

UFD iff ACCP and irreducibles are prime

Show that an integral domain $A$ is a unique factorization domain if and only if every ascending chain of principal ideals terminates, and every irreducible element of $A$ is prime.
2
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1answer
144 views

Subalgebra of free associative commutative algebra which is not free

I'm trying to provide a counterexample for analogue of Nielsen–Schreier theorem for the variety of associative commutative algebras (not necessary with unity) over a filed $F$. A counterexample for ...
5
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1answer
269 views

Antiderivative of Polynomials

I really like how differentiation is introduced for polynomials: Let $P(t) \in A[t]$ : $$D_P(t,s) = \frac{P(t) - P(s)}{t-s} \;\; \in A[t,s]$$ and the derivative of $P$ is $$P'(t) = D_P(t,t).$$ It ...
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2answers
549 views

No simple group of order $96$

I have to show that there is no simple group of order $96$ using the sylow theorems. I know that $96 = 2^5\cdot 3$, and from the third sylow theorem $n_2 = 1$ or $3$ and $n_3 = 1$ or $4$ or $16$. I ...
2
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2answers
391 views

Uniqueness of subgroups of a given order in a cyclic group

I am currently studying Serge Lang's book "Algebra", on page 25 it is proved that if $G$ is a cyclic group of order $n$, and if $d$ is a divisor of $n$, then there exists a unique subgroup $H$ of $G$ ...
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3answers
268 views

For any subgroup H of the group $G$, let $H^2$ denote the product $H^2=HH$. Prove that $H^2=H$.

For any subgroup of the group $G$, let $H^2$ denote the product $H^2=HH$. Prove that $H^2=H$. This question seems simple but I do not know how can I prove.
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0answers
66 views

Decompose a module $M$ of the form $N \times N$, where $N$ is simple

Let $M$ be a $\mathbb{C}[G]$-module of the form $M=N\times N$, where $N$ is simple. How to conclude that $M$ has infinitly many direct sum decompositions into two copies of $N$ ? This is what I have ...
2
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3answers
451 views

The ring $\{a+b\sqrt{2}\mid a,b\in\mathbb{Z}\}$

The set $\{a+b\sqrt{2}\mid a,b\in\mathbb{Z}\}$ spans a ring under real addition and multiplication. Which elements have multiplicative inverses? This is part of an exercise from an introductory text ...
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3answers
365 views

If A and B are ideals of a ring, show that A + B = $\{a+b|a \in A, b \in B\}$ is an ideal

If A and B are ideals of a ring, show that A + B = $\{a+b|a \in A, b \in B\}$ is an ideal I have the ideal test but no clue as to what to do with it: $a-b \in A$ whenever $a,b \in A$ ra and ar are ...
2
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1answer
84 views

Intermediate fields Separable, Algebraic, or Splitting

I just took my algebra final, and I had a few questions about intermediate fields and the properties of their extensions True or False...For $E \supset L \supset F$ $E \supset L$ is algebraic and ...
5
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2answers
231 views

$SU(2)$ Lie group

I have been studying Lie groups for a bit of fun for a while now and think they are fascinating. I have recently been told that $SU(2)$ can be used in some way to keep track of navigational systems in ...
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2answers
587 views

Why are modular lattices important?

A lattice $(L,\leq)$ is said to be modular when $$(\forall a,b\in L) x \leq b \implies x \vee (a \wedge b) = (x \vee a) \wedge b,$$ where $\vee$ is the join operation, and $\wedge$ is the meet ...