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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Proving $R=\{a+b\sqrt3\mid a,b \in \Bbb Z\}$ is Euclidean. [on hold]

Let $R=\{a+b\sqrt3\mid a,b \in \Bbb Z\}$ A. Prove $R$ is a Euclidian domain with respct to the norm $N(a+b\sqrt3)= |a^2-3b^2|$. B. Divide $1+\sqrt3$ by $2+\sqrt3$ in $R$.
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Is $2x^2+4$ reducible over $\mathbb C$?

I am not sure if I making some very fundamental mistake. But Gallian says that $2x^2+4$ is reducible over $\mathbb C$. If $D$ be an integral domain. A polynomial $f(x)$ from $D[x]$ is said to be ...
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May a monoid have two disjoint submonoids?

I'm asking this question inspired by the similar question about group and its subgroups. I tried to modify the proof presented there to work for monoids but I failed. I'm also not able to find any ...
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simple exercise in Cylindric algebra

I am trying to gain a better understanding of cylindric algebra, so I made up this example. Given a general rule that someone's father's father is his/her grandfather: $\forall_X ~ \forall_Y ~ ...
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2answers
87 views

Is an arbitrary group generated by a traversal of the conjugacy classes?

Let $G$ be a group, and let $\mathcal C$ be the collection of conjugacy classes of $G$. Let $S$ be a traversal of $\mathcal C$ (that is $S$ contains exactly one element from each set in $\cal C$). ...
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2answers
56 views

Proof for Surjections

I'm reading through Basic Algebra I (which I enjoy so far. Thoughts on this for self-studying?) and am having a difficult time proving surjection. I believe I understand the concept, but when it comes ...
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10 views

Simple $M_n(D)$-module with $D$ a division ring

Define $D$ to be a division algebra over a field $k$ and $R=M_n(D)$ the $n\times n$ matrix ring over $D$. A simple $R$-module $M$ is the quotient of $R$. I can write $R=\bigoplus_j I_j$ where $I_j$ is ...
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1answer
27 views

Quotient Gaussian Integers

Following Quotient ring of Gaussian integers, their extended conclusion is $\mathbb{Z}[i]/(a-ib) \cong \mathbb{Z}/(a^{2}+b^{2})\mathbb{Z}$. However it does not convince me, at least, one example ...
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1answer
17 views

Restricting Binary Operator $*$ To A Subset

I am currently reading Dummit and Foote's Abstract Algebra, and am having a little confusion over the following excerpt: Suppose that $*$ is a binary operation on a set $G$ and $H$ is a subset of ...
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44 views

Confused about proof in Basic abstract algebra by Bhattacharya

On page 264 , 2nd edition. Theorem 5.1 It says Let M be a free $R$-module with "a basis" $\{e_1,\dots,e_n\}$ Then $M$ is $R$-isomorphic to $R^n$. Above he is defining the standard basis as the ...
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Problems from Polynomial Rings. My attempt shown.

$1.$ Let $f(x) =a_mx^m+a_{m-1}x^{m-1}+ \cdots +a_0$ and $g(x) = b_nx^n+b_{n-1}x^{n-1}+ \cdots +b_0$ belong to $\mathbb Q[x]$ and suppose that $f \circ g \in \mathbb Z[x].$ Prove that $a_ib_j$ is an ...
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25 views

Text on Witt vectors that are accessible to undergraduate students

I am looking for a thorough text on Witt vectors that is accessible to an undergraduate student that have completed the following courses: Calc 1, 2, Linear Algebra and Abstract Algebra. (In Norway, ...
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1answer
46 views

Tensor product of duals

Can we show $V^* \otimes W^* \simeq (V \otimes W)^*$, when $V$ or $W$ is finite-dimensional without referring to the basis? I can inject $V^* \otimes W^*$ into $(V \otimes W)^*$ using the obvious ...
2
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1answer
41 views

Why are these dimensions equal?

For a finite $K$-algebra $A$ and $L\supset K$ fields, why do we have$$\dim_K A=\dim_L(A\otimes_KL)?$$ I ran across this a couple of times and it's always assumed to be quite obvious, which it isn't to ...
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19 views

AI / Machine Learning related to high/modern/front mathematics [on hold]

I major math and cs. and i'm interested in ai/machine learning/data mining. so i want to know what math subjects are used in frontier of these technology. especially, high mathematical tool, like ...
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1answer
15 views

When Is there no Local Power Integral Basis?

Let $A$ be a Dedekind domain, $K, L, B$ the usual designations of $A$'s quotient field, a finite separable extension of $K$, and integral closure of $A$ in $L$ respectively. If $\alpha \in B$ ...
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1answer
40 views

Proving result on algebraically closed fields

I have been told that: Let $f_1,\dots,f_n,g\in F[x_1,\dots,x_m]$ be polynomials in $m$ variables with coefficients in the algebraically closed field $F$. Then if the system: ...
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1answer
15 views

Proving a certain set is inductive

Let's give some context. We have to prove «Krull's theorem», which states that: If $A$ is a commutative ring, $N$ is its (nil)radical, i.e. the set of its nilpotent elements $\{x\in A:\exists ...
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4answers
263 views

Exercises in category theory for a non-working mathematican (undergrad)

I'm trying to learn category theory pretty much on my own (with some help from a professor). My main information source is the good old Categories for the working mathematician by Mac Lane. I find the ...
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R is a PID, and a is a nonzero nonunit in R. How can we show R/Ra is an injective module over R?

If we use Baer's criterion then it suffices to show that if there exist a map from an ideal $I$ to $R/Ra$ we must find a map $g$ such that $g\circ i=f$.
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2answers
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groups products vs vector space products

I started from wandering if the cross product (a product between two vectors that gives a vector) can be abstracted like dot product (a product between two vectors that gives a scalar) is abstracted ...
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1answer
17 views

proving closure of a subset [duplicate]

Let B be a set, and let * be a binary operation in B. Suppose * satisfies the associative law. Let $$P=\{b \in B : b * w = w * b \quad\forall\, w \in B\}$$ Prove that P is closed under *.
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Simple Cogenerator for the category of left $R$-modules

I want to prove that if the category of left $R$-modules has a simple cogenerator $M$, then $R$ is a simple ring. (Here, $R$ is an arbitrary ring with $1$.) My try is to take an ideal $I$ of $R$ ...
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1answer
26 views

Repeated Irreducible Representations in a representation

I'm reading through Serre's - Linear representations of finite groups. He has the following theorem (theorem 4 of chapter 2): Let $V$ be a linear representation of $G$, with character $\phi$ and ...
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1answer
34 views

Maps preserving roots of a polynomial function over finite fields

Let $P(x_{1},\ldots,x_{n}):\mathbb{F}_{2}^{n}\rightarrow \mathbb{F}_{2}$ be a polynomial function with degree $d$ and with variables $x_{1},\ldots,x_{n} \in \mathbb{F}_{2}$. Let $S(P)=\{ ...
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Can the derived subgroup be realized as an intersection of stabilizers?

For any group $G$, we have that $Z(G)=\bigcap_{x\in G}C_G(x)$ and that each $C_G(x)$ is the stabilizer of $x$ when $G$ acts on itself by conjugation. Is there a similar representation for $G'$? That ...
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1answer
42 views

Finitely generated, periodic group such that each conjugacy class is finite must be finite?

This is essentially a repeat of this question. However, the OP didn't seem to put in any work towards a solution and didn't provide context. Nevertheless, I'm still interested in the solution, and I ...
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1answer
25 views

Finitely generated ring of polynomials

Can we say that, "by definition", a ring $R[x]$ is finitely generated as an $R$-module for some commutative ring $R$ iff $x^n=q(x)$ for some polynomial $q(x)$ of degree $n-1$ for some $n$?
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number of group homomorphism from the symmetric group $S_3$ to the additive group $\Bbb Z/6\Bbb Z$ [duplicate]

The number of group homomorphism from the symmetric group $S_3$ to the additive group $\Bbb Z/6\Bbb Z$ ? a)1 b)2 c)3 d)0
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Number of normal subgroups of a non-abelian group of order 21 [on hold]

How many normal subgroups can a non-abelian group G of order 21 have other than the identity subgroup {e} and G? a)0 b)1 c)3 d)7
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1answer
52 views

If $F$ be a field, then $F[x]$ is a principal ideal domain. Does $F$ have to be necessarily a field?

If $F$ be a field, then $F[x]$ is a principal ideal domain. Does $F$ have to be necessarily a field? My Thoughts: Suppose instead of $F$, we take the set of polynomials $R[x]$ over a commutative ring ...
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“Type” in specifying a ring/ field

In the definition of fields on wikipedia, it says: A field is therefore an algebraic structure $\langle \Bbb F, +, \cdot , −, ^{−1}, 0, 1\rangle$; of type $\langle 2, 2, 1, 1, 0, 0\rangle$, ...
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1answer
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Suppose $x ≡ 1 \ (\text{mod}\ 8)$, and $x ≡ 9\ (\text{mod}\ 12)$ has a solution $x = x_0$. How many solutions modulo 24? [on hold]

Suppose $x ≡ 1\ (\text{mod}\ 8)$, and $x ≡ 9\ (\text{mod}\ 12)$ has a solution $x = x_0$. How many solutions modulo $24$ are there to this system of congruences?
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For m, n ≥ 2, let f : Z → Z/mZ×Z/nZ be the ring homomorphism defined by f(a) = (a + mZ, a + nZ)… read below. [on hold]

For m, n ≥ 2, let f : Z → Z/mZ×Z/nZ be the ring homomorphism defined by f(a) = (a + mZ, a + nZ). (i) The kernel K of f is the ideal qZ for some number q (depending, of course, on m and n). Describe q. ...
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55 views

Prove: A group $G$ is abelian if and only if the map $G\rightarrow G$ given by $x\mapsto x^{-1}$ is an automorphism.

Problem: A group $G$ is abelian if and only if the map $G\rightarrow G$ given by $x\mapsto x^{-1}$ is an automorphism. Prove it. Solution: $(\Rightarrow)$ If $f(x_1)=f(x_2)$, then ...
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1answer
51 views

If $G,H$ are monoids and not groups, prove that $f(e_G)=e_H$ may be wrong.

Problem: If $f:G\longrightarrow H$ is a homomorphism of groups, then $f(e_G)=e_H$ and $f(a^{-1})=f(a)^{-1}, \forall a\in G$. Show by example that the first conclusion may be false if $G, H$ are ...
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2answers
84 views

Existence of Irreducible polynomials over Z of any given degree which do not satisfy the Eisenstein's Criterion

I just came across the following interesting question which has been once discussed: Existence of Irreducible polynomials over $\mathbb{Z}$ of any given degree I was wondering if we could find such ...
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1answer
30 views

Global dimension regular rings of finite type

Have I made an error in my reasoning? If $k$ is a field, $A$ is a commutative regular $k$-algebra of finite type and ${\mathfrak{m}}$ is a maximal ideal in $A$ then since $Ext_{A_{\mathfrak{m}} ...
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31 views

A ring with prime characteristic

Let $p$ be a prime and let $R$ be a commutative ring with characteristic $p$. Prove that the number of elements of the set $$S_k=\{x\in R\;\lvert \;x^p=k\}\quad \text{for} \quad k\in ...
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1answer
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Open and closed equivalence relations

I am looking for canonical examples of open and closed equivalence relations, especially ones that are generated by a continuous functions. Intuitively I think that an open /closed continuous function ...
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Expansion of Cyclotomic polynomial on input $x+1$

I've been wondering why $$x^{p-1}+x^{p-2} + \dots + x + 1$$ expands to $$x^{p-1} + \binom p 1x^{p-2} + \binom p 2x^{p-3} + \dots + \binom p 1$$ when substituting $x$ for $x+1$ ? Can someone clarify ...
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1answer
54 views

Ring such that $q^2\mid p^2$ does not imply $q\mid p$?

Let $R$ be a commutative ring with $1$ and suppose $q^2\mid p^2,$ for $p,q \in R$. Unless $R$ is a UFD, I don't believe I can conclude that $q|p,$ but I would like to know a concrete counterexample. ...
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1answer
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Constructing an indicator function from a braid group which represents 'all strings have returned to their initial position'.

TL;DR Is there a well-defined closed formula from the braid group $B_n$ to $\{-1,1\} \left(\text{ or }\{0,1\}\right)$ which represents whether all the strings have returned to their initial ...
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1answer
37 views

Fixed Matrices over finite field by a map

Consider a set $M_n$ of all possible square matrices of dimension $n$ over a finite field $F_q$. Clearly the cardinality of the set $M_n$ is $q^{n^2}$. Let us consider a map $f:M_n$ $\longrightarrow$ ...
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Is there something that studies equivalent forms of writing and expression?

Supose we have: $x^2+x$, one could write it as $x(x+1)$ which would be equivalent to the first expression. I guess there might be a finite number of ways of writing expressions such that they are ...
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1answer
69 views

The Name for $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$ and $\mathbb{O}$ Exclusively Algebras?

The Wikipedia page for Normed Division Algebras has been redirected to Normed Algebras and the explanation given is that $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$ and $\mathbb{O}$ algebras are not the ...
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Linear combinations of roots of unity forming a commutative ring

How can it be shown that $$\mathbb{Z} [\zeta] = \{a + b\zeta^k \mid \zeta \text{ is a primitive $n$th root of unity; } 0 \le k \lt n ; \text{and } a,b \in \mathbb{Z} \} $$ is closed under addition? I ...
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Prove that a polynomial of degree $n$ over a commutative ring with zero divisors may have more than $n$ zeroes?

A polynomial of degree $n$ over a commutative ring with zero divisors may have more than $n$ zeroes. Attempt: Let $R$ be the commutatve ring which has a zero divisor $a \neq 0$. Then $\exists~~b \in ...
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Example of operation that is tree associative, but not generally associative

In a lot of algorithms using trees, we need the property that when folding $2^n$ elements with some operator $+$, we can do the first half of $2^{n-1}$ elements and the second half independently. Eg ...
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1answer
28 views

What are the elements of Z/10Z and of Z/2Z×Z/5Z, and identify which elements correspond under the map g from Z/10Z to Z/2Z × Z/5Z. [on hold]

What are the elements of Z/10Z and of Z/2Z×Z/5Z, and identify which elements correspond under the map g from Z/10Z to Z/2Z × Z/5Z. I know the elements of Z/10Z are {1,3,7,9}, is that the same for ...