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

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On the degree of cyclotomic fields extension.

Let $F$ be a field and $n$ be any integer satisfying $(n,\text{char}(F))=1$ when $\text{char}(F)\ne 0$. Let $\xi$ be a primitive $n$-th root of unity. I know that $\text{Gal}(F(\xi)/F)$ is isomorphic ...
2
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2answers
43 views

Proof of a property of set of all one-to-one mappings

Let $S$ be a nonempty set and $A(S)$ be the set of all one-to-one mappings of $S$ onto itself. I.N. Herstein in Topics in Algebra says (in page 28) that whenever $S$ has three or more elements, we can ...
2
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1answer
39 views

Determine the degree of an extension field over $\mathbb{Q}$

Let $\alpha = e^{\frac{i\pi}{6}}$. Compute $[\mathbb{Q}(\alpha):\mathbb{Q}]$ and find the minimal polynomial of $\alpha$, $m_{\mathbb{Q}}(\alpha)$. I can see clearly that $\alpha^6+1=0$ but I ...
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2answers
56 views

Proving that $f(x)$ is irreducible in $F[x]$ iff $\phi(f(x))$ is.

This is what I'm proving: Let $F$ be a field. Let $\phi : F[x]\to F[x]$ be an isomorphism such that $\phi(a)=a$ for every $a\in F$. Prove that $f(x)$ is irreducible in $F[x]$ iff $\phi(f(x))$ is. ...
1
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1answer
23 views

Applying hom-functor on short exact sequence

Let $N, M, M', M''$ be $R$-modules. Given homomorphisms $f: M' \rightarrow M$ $g: M \rightarrow M''$ $\psi: N \rightarrow M$ with $\operatorname{im} f = \ker g$, $g \circ \psi = 0$ and $g$ ...
4
votes
1answer
41 views

Brauer group of cyclic extension of the rationals

I am trying to compute the relative Brauer group of the cyclic Galois extension $L=\mathbb Q[x]/(x^3-3x+1)$ of $\mathbb Q$. I know that $$ \mathrm{Br}(L/\mathbb Q)\cong H^2(G,L^*)\cong\mathbb ...
0
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1answer
24 views

Good reference to study Free Algebras

I am interested in studying free algebras as Free Pre-Lie Algebras, Free Dendriform Algebras etc. But I dont know what a free algebra is in general. I found this definition on Wikipedia but it does ...
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1answer
21 views
0
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2answers
30 views

How do I get the Rational Canonical Form from the minimal and characteristic polynomials?

Let's say I have the minimal polynomial and characteristic polynomial of a matrix and all its invariant factor compositions. How do I get a rational canonical form matrix from this?
2
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2answers
32 views

Confusion about elements in fields, like -1 in Z5

I'm learning field and ring theory, and I've repeatedly seen the usage of -1, -2 and -3 as elements of $\mathbb{Z}_5$. As far as my knowledge goes, $\mathbb{Z}_5$ consists of {0,1,2,3,4}. This is ...
-1
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0answers
23 views

Difference between the set of generators and the alphabet of a free group

What do we mean by saying "a semigroup P is presented by generators and relations". Isn't it right only for the free semigroups? If it's right, we can't distinguish some two semigroups if they are ...
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0answers
33 views

Understanding isomorphism of Hom's

During some reading I came across the statement $$Hom_R(M,N) \otimes R/I \cong Hom_{R/I}(M/IM,N/IN) $$ where $M,N$ are $R$-modules and $I$ an ideal of the commutative ring $R$. This is proved, under ...
4
votes
1answer
60 views

$K/\mathbb{Q}$ contains a complex number - is complex conjugation in $\text{Aut}(K/\mathbb{Q})$?

Suppose $K$ is a finite extension of $\mathbb{Q}$. Suppose there is some complex number in $K$. Is it necessary that $\tau$, complex conjugation, is a member of $\text{Aut}(K/\mathbb{Q})$? I feel ...
0
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1answer
36 views

Degree of field extension over a fixed field

Suppose $L$ is a field and $H$ is a finite subgroup of $Aut(L)$. Then $[L:L^H]=|H|$ I've seen a proof of this that uses the Rank-Nullity theorem but it's rather cumbersome and difficult to remember ...
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2answers
67 views

Abstract Mathematics - Group theory and isomorphism

I have been trying to solve two problems, but I am stuck. Can anybody provide me with some links or theory to solve the following problems? The problems are from a study guide and the test exercises ...
0
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3answers
63 views

order $a$ = 5, $a^3b = ba^3$. show that that $ab = ba$.

Let $a, b$ be elements of a group $G$. Suppose that a has order $5$ and that $a^3b = ba^3$. I want to show that that $ab = ba$. Here is what I think: We know that we have $a^1, a^2, a^3, a^4, a^5 = ...
0
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2answers
35 views

Finding the parity of a permutation “exclusively”?

I'm trying to find the parity of permutations such as $(2468)$. What makes it possible to find the "exclusive" parity of such permutation? I.e. that if one tries to express $(2468)$ as a product of ...
0
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1answer
38 views

Commutativity of ring $R$ necessary for $\mathrm{Hom}_R(M,M')$ being an $R$-module

Why do we need $R$ to be commutative if we want $\mathrm{Hom}_R(M,M')$ (where $M$ and $M'$ are $R$-modules) to be an $R$-module itself? I tried to find out which axiom for modules does not hold if ...
0
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2answers
28 views

If $\sigma=(a_1 a_2 … a_n)$ and $|\sigma|$ is odd, then what is $\sigma^2$?

I'm trying to understand the way to infer the power of a permutation. If $\sigma=(a_1 a_2 ... a_n)$ is a $k$-cycle and $k=|\sigma|$ is odd, then how can I infer what $\sigma^2$ is?
0
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1answer
35 views

$x\in\overline{F}$ is in $F(\sqrt{F})$ $\iff$ $F\subset F(x)$ is finite Galois extension with Gal$(F(x)/F)$ abelian of exponent $2$

Let $F$ be a field of characteristic that is not $2$. I want to prove that $x\in\overline{F}$ is in $F(\sqrt{F})$ $\iff$ $F\subset F(x)$ is a finite Galois extension for which the group ...
0
votes
1answer
24 views

The question about second isomorphism theorem

If $G$ is a group and $N \trianglelefteq G$ and $K \leqslant G$, and $N_1 \leqslant N$, $N_1 \trianglelefteq G$, then can we say $NK/N_1 \approxeq K/K \cap N_1$?
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0answers
25 views

Maximum length of product of disjoint cycles?

This proof concerning the largest possible order of a permutation in $S_{10}$ uses some theorem for inferring that in $S_{10}$ the length $k_1+...+k_n$ of a product of disjoint permutations $k_1, ..., ...
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0answers
9 views

Submodules of direct sum of simple modules [duplicate]

Is it true that if $N$ is a simple module, then the only non trivial submodule of $N\oplus N$ is $N$? I am aware of the "diagonal" submodule $\{(x,x)\mid x\in N\}$, but that is isomorphic to $N$?
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0answers
31 views

Extension Operator.

I am working on my thesis about completion and extensions from an algebraic point of view. We have the closure operator which takes subsets to subsets with 3 criterias to meet $X\subseteq C(X)$ ...
0
votes
1answer
36 views

Are the free monoids always infinite?

It's the Wikipedia's definition of the free monoid: **In abstract algebra, the free monoid on a set is the monoid whose elements are all the finite sequences (or strings) of zero or more elements from ...
3
votes
3answers
42 views

How to proof equivalent condition of algebra morphism and coalgebra morphism about Hopf algebra

My question is why $\mu$ is multiplication preserving iff $\mu\circ m=(m\otimes m)\circ(1\otimes \tau \otimes1)\circ(\mu \otimes \mu)$ and this holds iff $m$ is comultiplication preserving, where ...
0
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0answers
81 views

Why is the Galois group of the polynomial $f(x)=x^5-x-1$ isomorphic to $S_5$?

Currently I am trying to understand why quintic (and higher degree) polynomials are not soluble by radicals, and one of the easiest examples of a quintic polynomial which has a nonsoluble Galois ...
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0answers
22 views

inverse of a root of f in an extension field K. [closed]

Let $f(x) = x^n + a_{n−1}x^ {n−1} + \cdots + a_0$ be an irreducible polynomial over $F$, and let $\alpha$ be a root of $f(x)$ in an extension field $K$. Determine the element $\alpha^{-1}$ explicitly ...
0
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2answers
39 views

Principal Ideal using coordinates?

I thought I understood principal ideals but now im stuck... I want to find the elements of the principal ideal $\langle(1,0)\rangle$ in the ring $\mathbb Z_3\times \mathbb Z_3$ with $+_3$ and $*_3$ in ...
0
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1answer
25 views

Basic quotient ring help

I am having a very hard understanding quotient rings. I never really understood quotient groups in abstract 1 but did well enough to get by. Now I am in abstract 2 and we hit quotient rings. I really ...
0
votes
2answers
23 views

Map of an element in a group to the conjugation by g

Let G be a group and suppose $g \in G$. $\varphi:G\rightarrow Aut\left ( G \right )$ $g \mapsto i_{g}$ is a Homomorphism with image $Inn\left ( G \right )$ where $Inn\left ( G \right )=\left \{ ...
4
votes
1answer
22 views

$G_{\mathfrak a}(A)$ integral domain and $\bigcap \mathfrak a^n = 0$ implies $A$ is integral domain

This is Lemma 11.23 in Atiyah: For an ideal $\mathfrak a \subseteq A$, define $G_{\mathfrak a} (A) = \bigoplus _{n=0} ^\infty \mathfrak a^n / \mathfrak a^{n+1}$. The statement of the Lemma: ...
2
votes
1answer
41 views

Showing the polynomials form a Gröbner basis

Let $A$ be an $m \times n$ real matrix in row echelon form and $I \subset \mathbb{R}[x_1,\dots,x_n]$ is an ideal generated by polynomials $p_i = \sum_{j = 1}^na_{ij}x_j$ with $1 \leq i \leq m$. ...
3
votes
1answer
60 views

Hilbert function and homogenous polynomials.

Let $\{[1:0:0],[0:1:0],[0:0:1],[1:1:1] \} = \{p_1,p_2,p_3,p_4\}$ be four points in the projective space $\mathbb{P}^2$. For every $p_i$, show there is a homogenous polynomial $f_i$ such that ...
0
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1answer
33 views

A division problem and its linearity of reminders

If the computation of remainder by division of $x_i$ by $y_1, \dots, y_n$ is $r_i$ for $i = 1,2$. Then for every scalars $c_1,c_2$, the remainder by division of $c_1x_1 + c_2x_2$ by $y_1,\dots,y_n$ is ...
0
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1answer
56 views

A group specified by Generators and Relations.

I'm confused with some terms in several definitions. Is an alphabet of a free group the same thing as a generator set of any group? If it is right, then by a given alphabet (set of generators) can be ...
4
votes
1answer
77 views

When is $\mathbb{Z} [x]/f(x) $ a Dedekind domain?

Given a monic separable irreducible polynomial $f$ with integer coefficients, when $\mathbb{Z} [x]/f(x)$ is a Dedekind domain? And when it happens to be a Dedekind domain, how to know its class ...
0
votes
1answer
21 views

Let $a^n, a^m \in (a^k)$ for some positive integer $k$. Then $k \mid n, m.$ Hence $k \mid \operatorname{gcd(n, m)}?$ [duplicate]

Let $a^n, a^m \in (a^k)$ for some positive integer $k$. Then $k \mid n, m.$ Hence $k \mid \operatorname{gcd(n, m)}$. How is it possible? Let $k = 6, n = 12, m =24.$ Context: Let $H$ be the ...
0
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1answer
31 views

How can we compute restrictions from a character table?

I would like to how to, when given a character table, calculate the restriction. $Res_H^G : Rep(G) \rightarrow Rep(H)$. For example: Let $G=S_4$ whose character table is given below (see ...
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0answers
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Kernel and Image of a group homomorphism defined by f(2) = 2 [duplicate]

Let $p$ be an odd prime. Let $f: \mathbb{Z}_{p^2} \to \mathbb{Z}_{2p}$ be the unique group homomorphism defined by $f(2) = 2$. Determine $ker(f)$ and $image(f)$. Help is very welcome. I am at a loss. ...
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2answers
36 views

Galois groups over p-adic fields

Let $k$ be a finite Galois extension of $\mathbb{Q}_p$. Do we know what groups $G$ show up as $G=Gal(k/\mathbb{Q}_p)$, as $p$ varies?
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64 views

Infinitely generated torsion free modules over PID

Let $R$ be a PID and $\mathbf{V}$ a torsion-free $R$-module, not necessarily finitely generated. If I understand it correctly, every rank 1 submodule of $\mathbf{V}$ is isomorphic to a submodule of ...
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0answers
15 views

units of the quotient ring of the integers over a prime power $[{\Bbb Z}/P^e\Bbb Z]^*$ is cyclic multiplicative group

I am studying Algebra as an extra curricular research project and in the reading I was assigned, the author somewhat offhandedly mentions that the units of ${\Bbb Z}/P^e\Bbb Z$, which is to say ...
0
votes
1answer
28 views

Determining automorphisms of this extension

I'm having a bit of trouble understanding an example from the introduction to galois theory section from Gallian's text. We have that $\omega = -1/2 + i\sqrt 3/2$ satisfies the equations $\omega^3 = ...
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0answers
9 views

Quotient Groups Composition Series Solvable [duplicate]

Show that all the quotient groups in a composition series of a finite solvable group G are cyclic of prime order. I know a polynomial equation is solvable in radicals if and only if its Galois group ...
1
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1answer
60 views

Prove that any ring consisting of 5 elements is either isomorphic to $\mathbb Z_5$ or has “null multiplication”

Prove that any ring $K$ consisting of 5 elements is either isomorphic to $\mathbb Z_5$ or has $ab = 0\ \forall a, b \in K$. Please help proving this. *** UPD Thank's to @almagest's hint I was ...
0
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1answer
21 views

Proving existence of automorphism

Let $G$ be a cyclic group, and $a,b$ two generators of $G$. Prove that exists automorphism $f$ such that $f(a)=b$ and that if $f$ is an automorphism then $f(a)$ generates $b$. $G$ is cyclic, thus ...
0
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1answer
39 views

Establish canonical isomorphism in Set category

Using notation $A^B := \mathsf{Mor}_{\mathcal{SET}}(B, A)$ establish canonical isomorphisms for any sets $X, Y$ and $Z$: $$ (Z^Y)^X \cong Z^{Y \times X} \; , \;(Z \times Y)^X \cong Z^X \times Y^X.$$ ...
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0answers
33 views

Intuition for Adeles and Ideles

I'm currently studying some class field theory and read about the notion of adeles and ideles. However, the object seems a bit arbitrary to me; is there a natural way to think about the adele-ring? ...
1
vote
1answer
38 views

Quaternion group and dihedral group.

The group $Q\subset GL_2(\mathbb{C})$ is generated by $\langle A,B\rangle$ $$ A= \left( \begin{array}{ccc}0 & 1 \\ -1 & 0\end{array} \right) B= \left( \begin{array}{ccc}0 & i \\ i & ...