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.

learn more… | top users | synonyms (1)

-2
votes
0answers
42 views

Extending our number system to include infinities [closed]

Instead of writing infinity using the infinity symbol, could we write such numbers as: |$\mathbb Z$| (size of the set of integer numbers) |$\mathbb R$| (size of the set of real numbers) Then ...
2
votes
1answer
72 views

Group isomorphism between $(\mathbb{Z}[x], +)$ and $(\mathbb{Q}^+, \cdot)$

I'm looking for a group isomorphism between the group of integer polynomials (with addition) and the group of positive rationals (with multiplication). I was thinking a map between generators of $\...
0
votes
0answers
27 views

Write $\mathbb{Z}^3/L$ as a direct sum of cyclic groups

Let $L\subset \mathbb{Z}$ be the subgroup of $\mathbb{Z}^3$ generated by the elements $(-1,-1,4),(2,4,0),(3,3,8)$. Write $\mathbb{Z}^3/L$ as a direct sum of cyclic groups. I've tried creating a ...
2
votes
0answers
58 views

Prime ideal of a polynomial ring in 6 variables

Let $k$ be a field and $k[x_1,x_2,x_3,y_1,y_2,y_3]$ a polynomial ring in 6 variables over $k$. How to prove that the ideal $(x_1y_2-x_2y_1,x_2y_3-x_3y_2,x_3y_1-x_1y_3)$ is prime in $k[x_1,x_2,x_3,y_1,...
1
vote
0answers
24 views

Separable but not reduced? [duplicate]

Say a $\Bbbk$-algebra is separable if $L\otimes _\Bbbk A$ is reduced for every field extension $L/\Bbbk$, and reduced if its underlying ring is reduced. Separable always implied reduced, and I found ...
1
vote
2answers
75 views

Prove every finite lattice has a greatest element - without induction

I have to prove that every finite lattice (L, ≤) has a greatest element. I have seen a lot of proofs proving this by using induction, however, I have to prove it without induction since our ...
2
votes
3answers
52 views

Problem in solving a question related to roots of an equation.

The question is : Show that the equation $x^n+x^{n-1}+\cdots+x-1=0$ has unique positive root for all $n \in \mathbb {N}$ and all these positive roots lying in between $0$ and $1$ for all $n \geq 2$...
4
votes
2answers
236 views

Given a finite Group G, with A, B subgroups prove the order of AB [closed]

How do you prove: Given a finite group $G$, with $A,B$ subgroups then $$|AB|=\frac{|A||B|}{|A \cap B|}.$$
1
vote
0answers
34 views

Finite dimensional separable algebra is étale

Say a $\Bbbk$-algebra is separable if $L\otimes _\Bbbk A$ is reduced for every extension $L/\Bbbk$. Say it's étale if there's an extension $L/\Bbbk$ such that $L\otimes_\Bbbk A\cong \prod_1^nL$. Here'...
1
vote
2answers
79 views

A finite abelian group containing a non-trivial subgroup which lies in every non-trivial subgroup is cyclic

Let $G$ be a finite abelian group s.t. it contains a subgroup $H_{0} \neq (e)$ which lies in every subgroup $H \neq (e) $. Prove that $G$ must be cyclic. Also what can be said about $o(G)$ ? I'm ...
2
votes
1answer
69 views

$A$ a subset of a finite group $G$ with strictly more than $|G|/2$ elements. Show $AA=G$. [closed]

The question asks (a) Let $A$ be a subset of finite group $G$ with strictly greater than $|G|/2$ elements. Show $AA=G$ and (b) Show this can fail in a monoid. I've been working on this for awhile ...
2
votes
0answers
63 views

On direct sum and direct product of groups

I understood the definitions of direct sums and direct products of groups (rings, modules, etc). The definitions of them can be given in terms of universal properties - certain commutative diagram in ...
0
votes
2answers
545 views

The elements of finite order in an abelian group form a subgroup: proof check

If G is an abelian group, show that the set of elements of finite order is a subgroup of G. Proof: Let G be an abelian group and H be the set of elements of finite order. (1) nonempty Now e ∈ H, ...
0
votes
3answers
67 views

An elementary proof that $k[x,y]/(xy-1)\cong k[x]_x$, where $k$ is a field

Letting $\phi:k[x,y]\to k[x]_x$, $\phi(x)=x$, $\phi(y)=\frac{1}{x}$, we see that $\ker \phi$ is prime, and $(1-xy)\subseteq\ker\phi$. Now, given that $k[x,y]$ has Krull dimension 2, $\ker\phi\neq (1-...
0
votes
0answers
27 views

Questions Concerning Proof of Artin-Rees Lemma

I have two questions about the proof of the Artin-Rees Lemma presented here: $\textit{Question 1:}$ Am I correct in assuming $I^0=R$? This is the only way some of the statements make sense I think. ...
1
vote
2answers
87 views

Group of order $1575$ problem

Problem Let $G$ be a group with $|G|=1575$. If $H \lhd G$ and $|H|=9$, then $H \subset Z(G)$. What I've done so far is $|G|=1575=3^25^27$. I consider $G$ acting on $H$ by conjugation, or, in other ...
-2
votes
1answer
100 views

What are some good books and materials for studying rings and fields theory? [closed]

I will very soon be introduced to the subject. I have heard this is one of the most important part of undergraduate algebra. I want to develop clear understanding in it from the beginning. I have ...
1
vote
1answer
521 views

In S4, find all the even permutation and show that the set of odd permutations isn't stable for binary operations in S4.

I want to find the even permutations of $S_4$ so i am supposed to find the transpositions right? but of what permutation exactly do i find the transpositions? And how do i know which ones are even? ...
3
votes
1answer
224 views

Semisimple objects in abelian categories

Let $\mathcal A$ be any Grothendieck abelian category and $0 \neq M \in \cal A$ an object. It is true that $M$ admits a simple subquotient? It is certainly true for $\mathcal A=R-Mod$ since $M$ ...
0
votes
1answer
17 views

Field structure of non solvable field extensions

I was considering the base field $D$ which is some solvable extenstion of $ \mathbb{Q}$, and a polynomial that isn't solvable in radicals such as $ x^5 - x + 1$. If we let $\zeta$ be a root of this ...
5
votes
3answers
177 views

Does $\gcd(a,bc)$ divides $\gcd(a, b)\gcd(a, c)$?

I want to prove that $\gcd(a,bc)$ divides $\gcd(a,b)\gcd(a,c)$ but I can't succeed. I tried to go with $\gcd(a,b) = sa+tb$ and it didn't work, tried to use the fact that $\gcd(a,b)$ and $\gcd (a,c)$ ...
1
vote
1answer
30 views

Confusion about generated subrings and subfields

In Milne's field theory notes he defines, given an extension $E/F$ and a subset $S\subset E$, the subfield of $E$ generated by $F$ and $S$ as the smallest subfield of $E$ containing both $F$ and $S$. ...
0
votes
0answers
39 views

Advantages and disadvantages of a particular definition of rings and subrings

My professor defines a ring as a triple $(A, +,\ \cdot)$ such that $(A, +, 0)$ is an abelian group, $(A,\ \cdot)$ is a semigroup and $\cdot$ distributes over $+$. Subsequently, $B\subseteq A$ is said ...
0
votes
0answers
48 views

Is it Noetherian and can you find an ideal which is non-finitely generated in it?

I have a lot of confusion in polynomial ring. Please help me explain it. Firstly, as we know, if $R$ is a ring then $R$ is sub-ring of $R[X]$ and $R[X]$ is a sub-ring of $R[[X]]$. I just wanna ask ...
3
votes
0answers
58 views

Does the uniqueness of splitting fields up to non-canonical isomorphism fall from algebraic geometry?

Does the uniqueness of splitting fields up to isomorphism over the base field fall out from some deep results in algebraic geometry, or is it something special to fields which I shouldn't expect to ...
1
vote
0answers
20 views

$f:A\to B$ flat homomorphism of rings, $Q$ is a prime ideal of $B,$ and $P=Q^c.$ Why is $B_Q$ a local ring of $B_P?$

I have a question about Exercise 2.18 on Introduction to Commutative Algebra by Atiyah and MacDonald. Let $f:A\to B$ be a flat homomorphism of rings, let $Q$ be a prime ideal of $B$ and let $P=Q^c,$ ...
-1
votes
1answer
25 views

the example of prime near-ring? [closed]

i need your help to find the example of prime near-ring, please. because i'm so confused to find the example of prime near-ring (the ring is near-ring, not a ring). I t
0
votes
0answers
64 views

Growth of the characters of finite permutation groups in the number of symbols

I have the following questions. When can the characters of the irreducible representations of the elements of a finite permutation group increase exponentially in the number of the symbols the ...
2
votes
1answer
51 views

Understand a part of the proof about permutations in a symmetric group on $n$ elements

Let $\sigma$ be an even permutation in $S_n$($\sigma \in A_n$). Assume $\sigma = \tau\sigma\tau^{-1}$ for some $\tau \in S_n$ and assume that the type of $\sigma$ consists of distinct odd integers. ...
1
vote
1answer
27 views

How to find a function/operator that satisfies the following conditions

I'm looking for a function that satisfies : 1) Symmetric: $f(x,y) = f(y,x)$ 2) Associative: $f(f(x,y), z) = f(x,f(y,z))$ 3) $f(x,x) = 0$ 4) it would be nice if $f(x,0) = x$, or at least that $g(x) ...
1
vote
0answers
17 views

$F(a_1,\ldots,a_n) = F(a_1+t_2 a_n+\cdots+t_n a_n)$ if Extension is Simple?

Suppose that F is an infinite field and that $L = F(a_1,\ldots,a_n)$ is a simple extension of $F$. Can you show, without the assumption that $[L:F] < \infty$ that $\exists t_2,\ldots,t_n \in F$ ...
3
votes
2answers
82 views

All ideals of a subring of $\Bbb Q$

Let $p$ be prime and $R=\{\frac{a}{b}:a,b \in \Bbb Z,b \neq0\text{ and }p\nmid b\}$. As an exercise, I have to prove that $R$ is a subring of $\Bbb Q$. My idea: With $a = 1$ and $b = 1$, $\frac{...
1
vote
2answers
51 views

Jacobson radical of $R=\{\frac{a}{b}:a,b \in \Bbb Z,b \neq0\text{ and }p\nmid b\}$.

I want to find the Jacobson radical, $J(R)$, of $R=\{\frac{a}{b}:a,b \in \Bbb Z,b \neq0\text{ and }p\nmid b\}$. Here my idea: One could use the definition $J(R)$ $=$ {$x \in R;\,\,\forall y \in R: ...
1
vote
1answer
35 views

Related to multiplicative subgroup of positive real line

Let $F$ be a subgroup of the multiplicative group $\mathbb R^*_{>0}$ such that $F$ is dense in $\mathbb R^*_{>0}$, $$N\cap F=\emptyset\ \text{ and }\ NF=N,$$ in which $N$ is a subset of $\...
-1
votes
0answers
66 views

Isomorphism between a quotient of a polynomial ring and a polynomial ring [on hold]

I'm going to show that $\mathbb{K}[x,y,z]/(y^2-xz)$ is not isomorphic to any polynomial ring. I'll be grateful of someone brings a hint to show this result. Thank you.
4
votes
1answer
105 views

Sum of free submodules of a module over a PID

It's possible to produce an example of an integral domain $R$ and a free $R$-module $M$ with free submodules $L, L'$ such that $L+L'$ is not free. We can take $R=M=K[x,y]$ , $L=\left<x\right>$ , ...
1
vote
1answer
296 views

The elements in the composite field $FK$

Where $F$,$K$ are two fields. What does the element in the composite field $FK$ look like? All the elements are generated by the elements of $F$ and $K$? (combination of the elements of $F$ and $K$) I ...
0
votes
1answer
21 views

If $G$ is a finitely presented group then is the commutator of $G$ isomorphic to the commutator of $F$ mod the relations?

Let $G=\langle\ S\ |\ R\ \rangle$ be a finitely presented group. Let $F$ be the free group with generating set $S$. Let $[F,F]$ and $[G,G]$ be the commutator subgroups of $F$ and $G$ respectively. Let ...
-1
votes
0answers
23 views

find a 2 sylow subgroup of $G=S_4\times S_3$ and a $3$ sylow subgroup. [closed]

how to find a $2$ sylow subgroup of $G=S_4\times S_3$ and a $3$ sylow subgroup. and the number of each number of subgroup.
14
votes
8answers
436 views

The proofs of the fundamental Theorem of Algebra [closed]

There are many proofs of the fundamental theorem of algebra. Which are the most beautiful proofs?
3
votes
1answer
64 views

How does Gauss's lemma follow from Nagata's lemma?

In section 4 of Samuel's Unique Factorization it's said Gauss' lemma is an easy consequence of Nagata's lemma. How does this work, i.e., how to deduce Gauss' lemma from Nagata's lemma? I'm asking ...
0
votes
0answers
28 views

Why $a+b$ is a generator of $F(a,b)$ over $F$, where $F$ is a field of characteristic zero.

Let $F$ be a field of characteristic zero. Assume that $a$ and $b$ are algebraic over $F$. The primitive element theorem says that there exists $w \in F(a,b)$ such that $F(a,b)=F(w)$; such $w$ is ...
2
votes
0answers
18 views

When the sum of two generators of a simple field extension is also a generator?

Let $F$ be a field of characteristic zero, and let $a$ be algebraic over $F$, so $K=F(a)$ is a finite separable field extension. Assume that $b$ is also a generator for $K$ over $F$, namely $K=F(b)$. ...
0
votes
1answer
13 views

If $B$ is a commutative domain, $Aut(B)$ acts on $Der(B)$ by conjugation

I'm reading Algebraic Theory of Locally Nilpotent Derivations by Gene Freudenberg, and I don't understand what's meant on the line $Aut(B)$ acts on $Der(B)$ by conjugation: $\alpha \cdot D = \alpha ...
1
vote
3answers
85 views

Let $a$ and $b$ belong to a group. If $|a|$ and $|b|$ are relatively prime, show that $\langle a \rangle \cap \langle b \rangle = \{e \}$

Let $a$ and $b$ belong to a group. If $|a|$ and $|b|$ are relatively prime, show that $\langle a \rangle \cap \langle b \rangle = \{e \}$. The most I can figure to do is to let $a^k = b^j$ for some $...
1
vote
1answer
29 views

A question about Ring theory

I'm studying basic Ring Theory. And in my textbook, the author states the definition of Euclidean domain: The integral domain $R$ is called to be a Euclidean domain precisely when there is a ...
0
votes
1answer
33 views

Group homomorphism in category theory [closed]

Too often i come across the statement a functor between two groups as categories is the homomorphism between the corresponding groups. This may be trivial, but has anybody proved is necessary and ...
1
vote
0answers
12 views

Linear forms on the abelian group of integer sequences

Let $S = \mathbf{Z}^\mathbf{N}$ be the abelian group of integer valued sequences. For each $i \in \mathbf{N}$ note $\delta_i\in S$ the sequence $(0,0,...,0,1,0,...)$ with the one at the $ith$ place. ...
2
votes
2answers
72 views

Number of irreducible polynomial over a field. [closed]

Find the number of irreducible monic polynomials of degree $2$ over a field with five elements. Please anyone help me.