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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Find a coset $f (= f + 22\Bbb Z)$ so that $(\Bbb Z/22\Bbb Z)^\times=U_{22} = \langle f\rangle$. [on hold]

Find a coset $[f]=f+22\Bbb Z\in \Bbb Z/22\Bbb Z$ such that the units, $(\Bbb Z/22\Bbb Z)^\times)=U_{22} = \langle f\rangle$ are generated by $f$. I am unsure of how to go about this.
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1answer
17 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
44 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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5answers
329 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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2answers
31 views

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
32 views

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
20 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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35 views

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 $R$ ...
0
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1answer
27 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
36 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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0answers
37 views

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
43 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
26 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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0answers
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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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0answers
27 views

Number of normal subgroups of a non-abelian group of order 21 [closed]

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
3
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1answer
55 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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27 views

“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
46 views

Suppose $x ≡ 1 \ (\text{mod}\ 8)$, and $x ≡ 9\ (\text{mod}\ 12)$ has a solution $x = x_0$. How many solutions modulo 24? [closed]

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. [closed]

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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1answer
62 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
55 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
95 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
33 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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2answers
39 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
30 views

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 ...
0
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2answers
37 views

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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2answers
96 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 $q\mid p,$ but I would like to know a concrete ...
2
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1answer
28 views

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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1answer
43 views

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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3answers
131 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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0answers
30 views

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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2answers
42 views

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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0answers
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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
31 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. [closed]

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 ...
2
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1answer
56 views

explicit matrix example of irreducible representation of s0(3)

Can someone give me a concrete or an explicit example of an irreducible representation of the Lie algebra so$(3)$? I know they are given by the Wigner D matrices but I want an explicit example of such ...
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1answer
38 views

Weibel “Introduction to homological algebra” Main Theorem 4.4.16

I can't understand the proof of Main Theorem 4.4.16 from Weibel's book "An Introduction to homological algebra". The Theorem states Let $R$ be a local noetherian commutative ring, then $R$ is ...
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Denesting radicals like $\sqrt[3]{\sqrt[3]{2} - 1}$

The following result discussed by Ramanujan is very famous: $$\sqrt[3]{\sqrt[3]{2} - 1} = \sqrt[3]{\frac{1}{9}} - \sqrt[3]{\frac{2}{9}} + \sqrt[3]{\frac{4}{9}}\tag {1}$$ and can be easily proved by ...
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4answers
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Finitely-generated group such that all (non-trivial) normal subgroups have finite index implies all (non-trivial) subgroups have finite index?

Let $G$ be a finitely generated group such that every non-trivial normal subgroup has finite index. Does it follow that every non-trivial subgroup of $G$ has finite index? This question arose as ...
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1answer
31 views

Irreducible Representations of $<X,Y>/\{[X,Y]=Y\}$

I was doing exercises from Etingof's Introduction to Representation Theory and came across this problem. $2.16.2$ Find all irreducible representations of the Lie algebra $L$ with generators $X$ and ...
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2answers
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Mistake in a question in Fulton's algebraic curves book?

I'm trying to solve this question in Fulton's book Algebraic Curves: I don't think this is true. Counter-example: $k=\mathbb R$, $n=1$ and $F=X_1^2+1$. Thanks
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4answers
160 views

Euclid's proof of the infinitude of primes to prove this question

I'm trying to prove that if $k$ is a field, then there are an infinite number of irreducible monic polynomials in $k[X]$. My attempt of solution is use almost the same strategy of the Euclid's proof ...
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1answer
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Show that if M, N are non-zero commutative rings, then M×N always has zero divisors, and is not an integral domain or a field.

Show that if M, N are non-zero commutative rings, then M×N always has zero divisors, and is not an integral domain or a field. How do I do this?!
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2answers
78 views

Let $f : \mathbb Z\to \mathbb Z/x\mathbb Z \times \mathbb Z/y\mathbb Z$ be the homomorphism defined by $f (n) = (n + xZ, n + yZ)$…

For $x,y \geq 2$, let $f : \mathbb Z\to \mathbb Z/x\mathbb Z \times \mathbb Z/y\mathbb Z$ be the ring homomorphism defined by $f (n) = (n + xZ, n + yZ)$. (i) The kernel $K$ of $f$ is the ideal ...
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0answers
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Can a ring of integers be free over a non-PID?

Let $K \subseteq L$ be an extension of number fields, and $A \subseteq B$ the corresponding rings of integers. $B$ is an $A$-module, generated by $[L : K]$ elements. If $K$ has class number one, ...
5
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1answer
63 views

Irreducibility of some multivariate polynomials

Consider the polynomials $xw-yz\in A[x,y,z,w]$ and $x^n+y^n+z^n\in A[x,y,z]$, where $A$ is a commutative ring. I am curious to know what conditions on $A$ (factorial ring, algebraically closed field, ...
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2answers
41 views

Prove that the field of quotients of an integral domain $D$ is the smallest field containing $D$. . My Attempt Shown

Let $D$ be an integral domain and let $F$ be the field of quotients of $D$. Show that if $E$ is any field that contains $D$, then, $E$ contains a sub field that is ring isomorphic to $F$. Hence, the ...
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1answer
26 views

Suppose that $n$ divides $m$ and that $a$ is an idempotent of $Z_n$. Show that the mapping $x \rightarrow ax$ is a ring homomorphism

Suppose that $n$ divides $m$ and that $a$ is an idempotent of $Z_n$. Show that the mapping $x \rightarrow ax$ is a ring homomorphism Attempt: Let $\Phi: Z_m \rightarrow Z_n$ be a ring homomorphism ...
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2answers
34 views

Let $F$ be a field. Show that the field of quotients of $F$ is ring isomorphic to $F$. My Attempt Shown

Let $F$ be a field. Show that the field of quotients of $F$ is ring isomorphic to $F$. Attempt: Let $F'$ be the field of Quotients of the field $F$. Let $\Phi:F \rightarrow F'$ such that ...