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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$\mathbb{Q}[\sqrt a_1,\ldots,\sqrt a_k]$ vs. $\mathbb{Q}(\sqrt a_1,\ldots,\sqrt a_k)$

Call an algebraic number polyquadratic if it can be expressed as the sum or difference of a finite number of square roots of rational numbers. (This definition follows Conway-Radin-Sadun rather than ...
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“adding” numbers $\in\mathbb{U\left(2\right)}\times \mathbb{R}^+_0$

Let a unitary number be one that corresponds to a matrix of the form: $$\left( \begin{array}{cc} w+i x & y+i z \\ -e^{i p} (y-i z) & e^{i p} (w-i x) \end{array} \right)$$ This is analogous to ...
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3answers
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Prove these subgroups are equal

I'm trying to solve this question which I found very complex and difficult due to the amount of details. Let $H, K, N$ be subgroups of a group G such that $H\leq K$, $H\cap N =K\cap N$, and ...
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1answer
570 views

Prove that the ring of formal power series over a field is an UFD

The problem is from Artin. Prove that the ring $\mathbb{R}[[t]]$ of formal power series given by $p(t)=a_0 + a_1 t+ a_2 t^2 + \cdots$ is an UFD. I have no idea how to do this. From the couple ...
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1answer
152 views

Help me understand: all finite separable field extensions are simple…

So I've been reading that this statement is true. Take, for example, $\mathbb{Q}(\sqrt[3]{2}, \omega)$ where $\omega$ is cube root of unity. This is the splitting field of $x^3 - 2$. It has degree ...
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2answers
721 views

How many unique Hamming (7,4,3) Codes are there?

I cannot find an answer to this on Google so I thought I would ask here. By unique I mean "distinct sets of codewords." By my count, there are $7!$ ways to choose an ordering of message bits and ...
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2answers
48 views

Number of homomorphisms from $\mathbb{Z}/4\mathbb{Z} \to \{\pm{1},\pm{i}\}$

I want to count the number of non-trivial group homomorphisms from $\mathbb{Z}/4\mathbb{Z} \to \{\pm{1},\pm{i}\}$. Since we want a homomorphism first thing is we want to map $\bar{0} \to 1$. Once ...
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103 views

Show that $A$ is a zero divisor of $S$ if and only if $A \neq 0$ and $\det(A)=0$.

Let $S=M_2(\mathbb{R})$. Show that $A$ is a zero divisor of $S$ if and only if $A \neq 0$ and $\det(A)=0$. Backward direction: Suppose $A \neq 0$ and $\det(A)=0$. Since $A$ is singular, in the ...
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68 views

Questions about representation theory of associative algebras.

I am reading the book Elements of the Representation Theory of Associative Algebras: Volume 1. I have two questions on page 85. On Line 18 of Page 85, it is said that $\ker p_i \subseteq ...
3
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1answer
231 views

Lagrange interpolation of Galois field functions

I'm trying to understand how to apply Lagrangian interpolation for a function $F : GF(2^k) \to GF(2^k)$, but am having some difficulty. Suppose I have a function $F : GF(2^2) \to GF(2^2)$ given by the ...
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299 views

$G$ is locally compact semitopological group. There exists a neighborhood $U$ of $1$ such that $\overline{UU}$ is compact.

Let $G$ be a group and $\mathcal T$ be a locally compact topology on $G$ such that for any $a\in G$ the functions $x \mapsto ax$ and $x\mapsto xa$ are continuous on $G$. How to prove elementarily ...
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3answers
227 views

Exhibit the correspondence in the Correspondence Theorem.

Here is the whole problem. I answered the first two parts, but I can't get down the third part. Problem Consider the group $D_{4} = \langle x,y:x^{2}=1, y^{4}=1, yx=xy^{3}\rangle$ and the ...
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2answers
172 views

Order-preserving isomorphism between $\mathbb{R}^n$ and $\mathbb{R}$

Suppose we have a linearly ordered group over $\mathbb Z^n$ where the ordering goes left-to-right, i.e. when deciding if $(x_1,x_2,\dots)<(y_1,y_2,\dots)$ we first check if $x_1< y_1$, if it is ...
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2answers
667 views

Determine Whether Two Matrix Groups Are Isomorphic Or Not.

Here is the problem I am having trouble with: Let: $$G = \left\{\begin{bmatrix} a & b\\ 0 & c \end{bmatrix} \in GL(2,R)\right\}$$ $$H = \left\{\begin{bmatrix} a & 0\\ 0 & b ...
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1answer
202 views

Using roots of unity to find all the roots of a given polynomial

Assume we're working in $\mathbb{Q}[x]$. So I know that for a polynomial like $f(x) = x^3 - 2$, the roots are found by realizing that $\sqrt[3]{2}$ is obviously a root, and then having the other 2 ...
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3answers
107 views

Order of matrices in $SL_2({\mathbb{F}_q})$

Could you tell me how to prove that in $SL_2({\mathbb{F}_q})$ the only element of even order is $-I$ ($ \ I$ - identity matrix)? I would really appreciate a thorough explanation, because I cannot ...
2
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1answer
204 views

Group theory - proof check about index and quotient group

I'm studying Cayley's Theorem on the Humpreys "A Course in Group Theory" and i did not understand a passage in a preposition. (pag 86 Corollary 9.23). It claims: "Let $H \leq G$ with finite index $n$. ...
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2answers
131 views

Number of Abelian Groups of Order 256

I am trying to find the number of abelian groups of order 256. Is the following correct? We may write $256=2^8$ we then know that this may be represented in the form: $C_{n_1}\times.....\times ...
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1answer
480 views

How are binary operations used in the real world?

Not necessarily a mathematical question, but how could binary operations be used in the real world? What applies to it?
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97 views

Showing that if $N \le G$ is finite minimal normal with every simple homomorphic image abelian, then it is abelian itself

I've been working on the following problem, with no success so far: "Let $N$ be a finite minimal normal subgroup of a group $G$, and suppose $N$ has the property that every simple homomorphic image ...
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1answer
287 views

Field of fractions of $\mathbb{Q}[x,y]/\langle x^2+y^2-1\rangle$ [duplicate]

This problem goes as follows: Prove that $\mathbb{Q}[x,y]/\langle x^2+y^2-1\rangle$ is an integral domain and that its field of fractions is isomorphic to the ring of rational functions ...
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67 views

Find with proof the number of units in the ring $R=\mathbb{Z}_8 \times \mathbb{Z}_9 \times \mathbb{Z}_5$

Find with proof the number of units in the ring $$R=\mathbb{Z}_8 \times \mathbb{Z}_9 \times \mathbb{Z}_5$$ Since $\gcd(5,8,9)=1$, by Chinese Remainder Theorem, I get $R=\mathbb{Z}_{360}$ Since ...
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5answers
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Definition of <presentation> of a group

basing myself on suggestions I found in previous discussions, I have opted for the algebra book of Dummit. However, I have now a little problem. On page 26 (3th edition) the authors say that ...
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1answer
48 views

$HK\cap N=H(K\cap N)$

I'm trying to prove If $H$, $K$ and $N$ are subgroups of a group $G$ such that $H\lt N$, then $HK\cap N=H(K\cap N).$ I'm trying sets inclusion to prove it, am I in the right way? I need help. Thanks ...
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3answers
435 views

Ideal generated by a set in a commutative ring without unity

In a commutative ring with unity $1$, call it $R$, the the ideal generated by the set $S=\{a_1,...,a_n\}$ is the smallest ideal of $R$ containing $S$. It can be proven that this ideal is $$ ...
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2answers
80 views

Conjugation in $S_n$

We have that any element in $S_n$ is generated by the adjacent transpositions $(12),\dots,(n-1,n)$. I am trying to calculate the conjugation of $(ab)\in S_n$ by $\sigma\in S_n$. So if we write sigma ...
6
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1answer
209 views

Dirichlet's theorem on arithmetic progressions

The theorem can be found on Wikipedia. In the subsection "Proof" Wikipedia says that there is a proof for the case $a=1$ which uses no calculus, instead splitting behavior of primes in cyclotomic ...
6
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1answer
159 views

A question on an answer on Math Overflow about Artin approximation

I have a question on an answer of this Math Overflow question. Let $(A,I)$ be a commutative excellent normal local domain. The completion $$ \hat A=\underset{\longleftarrow}{\operatorname{lim}} ...
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4answers
1k views

Finite fields and primitive elements

Let $\mathbb F_9$ be a finite field of size $9$ obtained via the irreducible polynomial $x^2 + 1$ over the base field $\mathbb F_3$. How can you find a primitive element? Make a list of the ...
5
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1answer
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How to show polynomial is irreducible in extension field of rationals?

How would one show this? is there any test? say $x^3 - 2x - 5$ over $\mathbb{Q}[\sqrt{2}]$ This is just a made-up example, but I'm interested in the general process and how it differs from showing ...
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0answers
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Description of $\mathrm{Ext}^1(R/I,R/J)$

Let $R$ be a commutative ring with unit and $I$ and $J$ are nonzero ideals of $R$. Do we have a nice description for $\mathrm{Ext}^1_R(R/I,R/J)$? What do I mean by a nice description? For example ...
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Exceptional Simple Jordan Algebra Cross product

Does anyone happen to know of an explicit construction of the cross product on the exceptional simple Jordan algebra, or perhaps a reference? Context: I'm trying to see if $(D^* a)X b + a X (D^*b) = ...
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1answer
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Noetherian and Artinian modules over subrings

I have a question about whether Noetherian-ness and Artinian-ness of modules are preserved under changes of the base ring. More precisely: Let $R$ be a commutative ring and $S \subseteq R$ a ...
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2answers
329 views

Quotient rings of Gaussian integers

I used this isomorphism today but now I'm having trouble justifying it. The norm function isn't additive so I can't come up with a ring isomorphism to prove the following: For any $\,a+bi\in\Bbb ...
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Show that a set is denumerable.

I have the following question: Let $A=\{x:\exists m,n \in\mathbb{Z} \text{ such that } x=m+n\sqrt{p}\}$, where $p\in\mathbb{Z}$ is a fixed prime. Show that $A$ is denumerable. I hope someone can ...
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1answer
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Characters of elements under every representation equal implies conjugacy

If $G$ is a group, suppose that for every $G$-module $V$ we have $$\chi_V(g_1)=\chi_V(g_2).$$ How can I be sure $g_1$ and $g_2$ are conjugate in $G$? Its easy to the reverse implication; ...
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If $[G:H]$ and $[G:K]$ are relatively prime, then $G=HK$

I'm struggling to proof that if $H$ and $K$ are subgroups of a finite index of a group $G$ such that [G:H] and [G:K] are relatively prime, then $G=HK$. I don't know why I can't answer it, because this ...
4
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3answers
378 views

finite abelian group satisfying $x^2=e$

I looked but didn't see this question pop up. Not homework as I am graduating on Thursday and took Abstract a year ago. I'm taking the Praxis II and honing my skills. I have good intuition about ...
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1answer
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Existence Proof: $T(v_i)=w_i$ for all $i=1,2,3,\dots,n$

Theorem to prove: Let $\{v_1,\dots,v_n\}$ be a linearly independent set in a finite-dimensional vector space $V$ and let $w_1,\dots,w_n$ be arbitrary vectors in a vector space $W$. Then there exists ...
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Revisited: $[T]^{\gamma}_{\beta}$ is diagonal?

Let $V$ and $W$ be finite-dimensional vector spaces with $\dim(V)=\dim(W)$ and let $T:V \rightarrow W$ be a linear map. How do I prove that there are bases $\beta$ of $V$ and $\gamma$ of $W$ such that ...
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Confusion regarding the augmentation map of Lie algebras

Let $\mathfrak{g}$ denote a Lie algebra, $K$ a field, and view $K$ as a trivial $\mathfrak{g}$-module (that is, define $x \cdot a = 0$ for all $x \in \mathfrak{g}, a \in K$). In other words, we have a ...
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1answer
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A problem from Gallian's Algebraic Extensions

Let $a$ be a complex number that is algebraic over $\mathbb{Q}$ and let $p(x)$ denote the minimal polynomial of $a$ over $\mathbb{Q}$. Show that $\sqrt{a}$ is algebraic over $\mathbb{Q}$ and ...
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3answers
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If $ G $ has no non-trivial automorphism, then $ G $ is abelian and $ g^2 = e $ for all $ g \in G $ .

If $ G $ has no non-trivial automorphism, then $ G $ is abelian and $ g^2 = e $ for all $ g \in G $ . With the assumption, I dont know how to start the proof. If there is no non-trivial ...
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3answers
400 views

All group homomorphism from $ \mathbb{Z} _m $ to $\mathbb{Z}_n $ [duplicate]

All group homomorphism from $ \mathbb{Z} _m $ to $ \mathbb{Z}_n $ How could I find every group homomorphism?
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1answer
981 views

A non-abelian group of order $ 6 $ is isomorphic to $ S_3 $

I know that it is duplicated. But I'm confusing some step of this proof. Please help me. pf) Let $ G $ be a nontrivial group of order $ 6 $. Since $ G $ is non-abelian, no elements in $ G $ have the ...
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Injective and Projective module

Ok, this problem is driving me nuts. At first, I thought I did it. But when reading another textbook (having a similar proposition, they (the problem in my textbook, and the proposition in the other ...
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1answer
109 views

Does someone know other examples of $K$-semigroups?

In our work on projective representations we need use the following object: Let $K$ be a field. By a $K$-semigroup we mean a semigroup $S$ with $0$ and a map $K \times S \to S$ such that $\alpha ...
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If $G$ is a group, $H,K \leq G$, $K \subseteq H$, $\left[G:H\right]$ and $\left[H:K\right]$ both finite then… [duplicate]

If $G$ is a group, $H$ and $K$ both subgroups of $G$, $K \subseteq H$, $\left[G:H\right]$ and $\left[H:K\right]$ both finite then $\left[G:K\right]=\left[G:H\right]\cdot\left[ H:K \right].$ I am ...
4
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1answer
94 views

let $f (x) = x^p - a \in F[x]$. Show that $f (x)$ is irreducible over $F$ or $f (x)$ splits in $F$.

Let $F $ be a field of characteristic $p$ and let $f (x) = x^p - a \in F[x]$. Show that $f (x)$ is irreducible over $F$ or $f (x)$ splits in $F$. I am completely stuck on it.can someone help me ...
3
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1answer
202 views

Let $F = \mathbb{Q}(\pi^3)$. Find a basis for $F(\pi)$ over $F$.

Let $F = \mathbb{Q}(\pi^3)$. Find a basis for $F(\pi)$ over $F$. How can I solve this. Can anyone help me please. Thanks.