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.

learn more… | top users | synonyms (1)

1
vote
0answers
12 views

Proving a subgroup is normal

Problem Let $G$ be a group with $|G|=pm$, $p$ prime and $p \geq m$. Suppose there is $H$ subgroup of $G$ with $[G:H]=p$. Show that $H$ is normal. This problem was given to me in class just after ...
1
vote
1answer
12 views

Find the splitting field of a polynomial

The extension $\mathbb{Z}_p \leq \mathbb{F}_{p^n}$ is normal, as the splitting field of the polynomial $f(x)=x^{p^n}-x$ ($\mathbb{Z}_p$ is a perfect field therefore each polynomial is separable). So, ...
1
vote
1answer
11 views

Show sentences-Field of rational functions

Let $K=\mathbb{Z}_p(x,y)$ be the field of rational functions of variables $x,y$ with coefficients in the field $\mathbb{Z}_p$, where $p$ is prime. Let $g(t)=t^p-x, h(t)=t^p-y \in K[t]$ and $E$ is the ...
-1
votes
2answers
12 views

$\mathbb Z_{p^2}$ is not a non trivial semidirect product.

I am trying to prove that the group $\mathbb Z_{p^2}$ (p prime) is not a non trivial semidirect product. Since a group $G \cong K \rtimes H$ if and only if for all short exact sequences $$0 ...
1
vote
2answers
21 views

Showing that the product group of $G$ and $H$ satisfies the universal property for coproducts in the category of abelian groups $\mathbf{Ab}$

I'm working on another problem of Aluffi's Algebra. Given the category $\mathbf{Ab}$ of abelian groups, the problem is to show that for any two groups $G$ and $H$ the product group $G\times H$ ...
2
votes
2answers
41 views

For any group $G$, $|G/Z(G)| \neq 91$.

In Malik's Fundamentals of abstract algebra, one can find the following problem: Prove that for any group $G$, $\vert G/Z(G)\vert \neq 91$. This exercise is just ahead of Sylow's theorems. I've ...
0
votes
0answers
26 views

Show that the fields are equal

I have to show that $\mathbb{F}_{2^2}=\mathbb{Z}_2(a)$, where $a \in \mathbb{F}_{2^2}$ is of degree $2$ over $\mathbb{Z}_2$. $$$$ To show this do I have to take first an element of ...
3
votes
1answer
24 views

Direct sum of simple modules and Schur's Lemma

Suppose $M,N$ are two non-isomorphic simple $R$-modules. For $m,n\geq1$, is it true that $$ \text{Hom}_R(M^{\oplus m},N^{\oplus n})\cong\hat{0}\,? $$ I think it's true by Schur's Lemma. ...
0
votes
2answers
26 views

Existence of a nontrivial solution to a polynomial equation

Let $p \ne 0$ and consider the equation $$ x_1 (x_1 + p)^2 + \dots + x_n (x_n + p)^2 = 0.$$ Does there exist a solution $x \in \mathbb R^n$ to this equation that is not the trivial solution $x=0$?
2
votes
2answers
38 views

Does (Z, +) have two generators but infinitely many generating sets?

We say the group of integers under addition Z has only two generators, namely 1 and -1. However, Z can also be generated by any set of 'relatively prime' integers. (Integers having gcd 1). I have ...
0
votes
1answer
28 views

What am I not understanding about the canonical proof that no finite field is algebraically closed?

Wikipedia gives what seems to be the canonical proof that no finite field is algebraically closed, under the article titled “Algebraically closed field”. It gives the following definition: “In ...
3
votes
1answer
38 views

Commutators in a group

Let $G$ be a group and for $x,y\in G$, define $[x,y]=x^{-1}y^{-1}xy$ to be the commutator of $x$ and $y$. If $y_1,\cdots,y_n\in G$, is it true that $[x,y_1\cdots y_n]$ can be written of a product of ...
2
votes
3answers
164 views

Degree of field extension is infinite

If we have the field extension $\mathbb{Q}\leq \mathbb{R}$, could you explain me why it stands that $[\mathbb{R}:\mathbb{Q}]=+\infty$ ??
2
votes
1answer
23 views

Multiplicative group of a field contains maximal n-1 elements with order n

Let $F$ be a field and $n\in \mathbb N,n>1$. I want to show that the multiplicative group $K$\ $\{0\}$ contains maximal $n-1$ elements with order $n$. I actually don't have any ideas how to solve ...
1
vote
1answer
32 views

A finite-dimensional $K$-algebra?

Let $A$ be a commutative noetherian domain of characteristic $p>0$ and $K$ be its quotient field. Let $G$ be any finite group whose order is divisible by $p$. Is $KG$ a finite-dimensional ...
1
vote
0answers
33 views

Automorphisms of $Z_{p^{i_1}}*Z_{p^{i_2}}*…*Z_{p^{i_n}}$

If $Z_{p^{i_1}}\times Z_{p^{i_2}}\times\cdots\times Z_{p^{i_n}}=\langle a_1,...,a_n\rangle$, then each automorphism of this group is the forms as follows, $$\sigma:a_j\rightarrow ...
2
votes
4answers
273 views

Can an empty set be both torsion and torsion free group?

I was wondering if an empty set can be a torsion group (since the definition of torsion group is that if $x$ is in the set $X$ has a finite order. However, the assumption is false, so the implication ...
1
vote
0answers
19 views

A question on vectors represented by multilinear polynomials

Let the set of multilinear polynomials of degree atmost $t$ in $\Bbb Z[x_1,\dots,x_n]$ be $\Bbb Z^{t}[x_1,\dots,x_n]$. Let $S=\{0,1\}^n$. Fix an ordering of $S$. For every $f\in\Bbb ...
0
votes
0answers
15 views

Similarities/differences between multivariate polynomials and integers

There are a few questions on this site that asks for similarities between integers and univariate polynomials. I am wondering if multivariate polynomials have any related analogies with integers.
1
vote
1answer
24 views

Linearly equivalent divisors and linear transformations

Let $C$ be a projective nonsingular irreducible curve. Let $D$ be a divisor on $C$. Suppose $l(D) = n > 0$ and $L(D) = \langle f_1, \ldots, f_n \rangle$. Consider the map $\varphi_D : C \to ...
0
votes
0answers
9 views

Prove that $e$ is a neutral element to the left of $*$ iff $Lc\equiv Tc$.

I have the following exercise to do: Given a magma $(C,*)$, for each $c\in C$ we have the functions: $\begin{matrix}{Lc:C\to C}\\{a\mapsto lc(a):=c*a} \end{matrix}$ ...
0
votes
1answer
22 views

Galois group of a polynomial and subfields

Suppose $f(x)\in \mathbb{Z}[x]$ is an irreducible quartic whose splitting field has Galois group $S_4$ over $\mathbb{Q}$. Let $\theta$ be a root of $f(x)$ and set $K=\mathbb{Q}(\theta)$. a) Prove ...
0
votes
1answer
14 views

Computing Cosets and the Kernel

If we let $h:\Bbb{Z}_3\to \Bbb{Z}_3 \times\Bbb{Z}_6$ where $h(a) = (a,2a)$ What is the kernel and how can cosets be defined on this function? Any assistance or solutions are welcome.
0
votes
2answers
37 views

Looking for help, Aluffi Exercise 5.13, Chapter 6: characterization of PIDs

I quote: "Let $M$ be a finitely generated module over an integral domain $R$. Prove that if $R$ is a PID, then $M$ is torsion-free if and only if it is free. Prove that this property characterizes ...
1
vote
1answer
32 views

Show that $x\otimes x+2\otimes2$ is not an elementary tensor in $I\otimes_{A}I$ [duplicate]

Let $A=\mathbb{Z}[x]$ and $I=(2,x)\lhd A.$ Show that $x\otimes x+2\otimes2$ is not an elementary tensor in $I\otimes_{A}I$. I have $$I\otimes_A I = \frac{L_A(I\times I)}{T}$$ where $T$ is the ...
0
votes
1answer
12 views

$M_n(D)$ is left and right-simple?

Is it true that if $D$ is a division ring and $n\in\mathbb{Z}_{\geq1}$, then the only left and right ideals of the ring $M_n(D)$ are the trivial ones? I know that $M_n(D)$ is simple, and the ...
0
votes
0answers
14 views

dimension of vector space $\frac{\langle e_{ab_1\ldots b_p}\rangle}{\langle \sum_{1\leq i\leq p}e_{ab_1\ldots \widehat{b_i}\ldots b_pc}\rangle}$

Let $p$ be a prime and $n\!\in\!\mathbb{N}$. What is the dimension of the $\mathbb{Z}_p$-module $$V_{p,n}=\frac{\langle e_{ab_1\ldots b_p};\: 1\leq a<b_1<\ldots<b_p\leq n\rangle}{\langle ...
1
vote
0answers
31 views

Can someone explain me about Sylow 2-subgroups in S3 x S3?

I found this explanation in a Stanford webpage. However, I am not clear how to find the 9 Sylow 2-subgroups. Can someone explain me? Thanks!
1
vote
1answer
23 views

Derived subgroup of a finite non-Abelian p-group is proper?

How do I show that the derived subgroup of a finite p-group is always proper? In Abelian groups, it's trivial. In non-Abelian groups, my intuition is that there should be some way to relate G/Z(G) to ...
0
votes
1answer
42 views

Steps to construct the Field of fractions of Gaussian Integers $\mathbb{Z}[i]$ [duplicate]

i don't know how to construct such field $\mathbb{Q[i]} $ from $\mathbb{Z[i]}$. I know the following: $(a+bi,c+di)\sim (m+ni,r+si)$ iff $(a+bi)(r+si)=(c+di)(m+ni)$ is the equivalence relation and if ...
4
votes
1answer
32 views

Existence of projectives in the category of torsion abelian groups

Consider the category of torsion abelian groups. This category doesn't have enough projectives by the following argument. Suppose $C_2$ (cyclic group of order 2) is the homomorphic image of a ...
0
votes
1answer
23 views

Quotient Ring and finite fields

How is a quotient ring $\mathbb Z/p^e\mathbb Z$ (where p is prime and $e>2$) different from a finite field $\mathbb F_{p^e}$? When they are both rings, have the same elements? I thought a finite ...
1
vote
2answers
30 views

Show that an extension is separable

Let $K$ be a field with $\operatorname{char} K=p$, where $p$ is a prime, and let the degree of the extension $K \leq L$ be coprime to $p$. How can I show that the extension is separable?? Could you ...
1
vote
1answer
35 views

Atiyah & MacDonald on local Noetherian and Artinian rings - sanity check.

In the chapter on Artinian rings in "Introduction to Commutative Algebra" by Atiyah and MacDonald, we have: Proposition 8.6. Let $(A,\mathfrak{m})$ be a local Noetherian ring. Then exactly one of the ...
1
vote
1answer
23 views

Correspondence between ideals of $R$ and $D^{-1}R$

Let $R$ be an integral domain, and $D\subset R$ be a multiplicatively closed subset such that $1\in D$ and $0\not\in D$ . Prove/disprove that there is a one-to-one correspondence between the ideals of ...
1
vote
1answer
29 views

A subgroup of $\mathbb{Z}$

Let $A$ be a subgroup of $\mathbb{Z}$. Show that $A=\{0\}$ or $A\cong\mathbb{Z}$. My intuition is to do something with the generator(s) of A (maybe it's not the best thought), but I have no idea how ...
1
vote
2answers
34 views

Trouble on Aluffi's Exercise 5.5, Chapter 6: finitely generated projective module over a local ring is free

The overall goal of the problem is the following: Let $R$ be a local commutative ring (so that there is a unique maximal ideal $\mathfrak{m}$) and let $M$ be a finitely generated $R$-module. Assume ...
0
votes
1answer
17 views

Prove that $\Bbb F_p^\times$ is equal to Miller–Rabin primality test for prime number

I want to prove, that $\Bbb F_p^\times = MRP(p)$. I think, that I have to start with this statement: $\{a \in \Bbb F_p^\times | a^2 = 1 \} = \{1; -1\}$ But I do not know how to continue this idea.
2
votes
1answer
24 views

Suppose that $R$ is a commutative ring with unity, $a \in R$, and $\varphi(r) = ar$ defines a ring homomorphism. Prove that $a$ is idempotent.

Suppose that $R$ is a commutative ring with unity, $a \in R$, and $\varphi(r) = ar$ defines a ring homomorphism $\varphi$. Prove that $a$ is idempotent, i.e. that $a = a^{2}$. This is exercise 15 ...
1
vote
1answer
32 views

If $\lvert g \rvert=m$ is finite then prove that $ng=0$ if and only if $m\mid n$.

Let $G$ be an abelian group and let $g \in G$. If $\lvert g \rvert=m$ is finite then prove that, for $n\in \mathbb Z$, $ng=0$ if and only if $m\mid n$. I think this amounts to proving that: $$ng=0 ...
1
vote
2answers
33 views

Complex coefficients in a quadratic equation [on hold]

Can the general solution to the quadratic equation be used if the coefficients are complex?
4
votes
2answers
52 views

Polynomials over $\mathbb{F}_2$ without multiplicity 1 factors

Let $f \in \mathbb{F}_2[T]$ such that both $f$ and $f + 1$ have the property that every irreducible factor in the unique factorization domain $\mathbb{F}_2[T]$ appears with multiplicity at least $2$. ...
0
votes
1answer
21 views

Calculation of the Cosets of the Kernel

Let $g:\Bbb{Z}_3 \times\Bbb{Z}_4 \to \Bbb{Z}_3$ where $g((a,b)) =a$ . What are the cosets of the kernel? I understand that a kernel is the set of elements that map to the identity element. Any ...
1
vote
2answers
44 views

Difference between Integral Domains and Fields.

Can someone please help me in figuring out how all fields are integral domains but not all ID are fields? My course assumes IDs to be commutative with unity but fields require all elements to have a ...
1
vote
2answers
28 views

Let $F$ and $K$ be fields. If $F\subseteq K$ and $r\in K$ s.t. $r^2$ is algebraic over $F$. Then $r$ is algebraic over $F$.

Let $F$ and $K$ be fields. If $F\subseteq K$ and $r\in K$ s.t. $r^2$ is algebraic over $F$. Then $r$ is algebraic over $F$. Assume polynomial $p(x)\in F[x]$ s.t. $p(r^2)=0$ If $r\in K$ and $r^2$ is ...
0
votes
1answer
24 views

Finding the kernel of $\phi:\Bbb{Z}_6\to\Bbb{Z}_2 $ given by $\phi(x)=$ the remainder of $x$ when divided by $2$

Let $\phi:\Bbb{Z}_6\to\Bbb{Z}_2 $ be given by $\phi(x)=$ the remainder of $x$ when divided by $2$. I have become fairly confident with calculating the left and right cosets, but what is the kernel ...
2
votes
1answer
40 views

What is the name of this terminology?

Let $G$ be the group generated by a set $X=\{x_1,\cdots,x_n\}$. Then each element can be (not necessarily uniquely) written as a product of the form $x_{j_1}^{e_1}\cdots x_{j_k}^{e_k}$, where each ...
1
vote
2answers
25 views

How to show that an ideal of $F[x]$ containing an irreducible polynomial of degree $n$ and a nonzero polynomial of degree $<n$ is $F[x]$?

Let $F$ be a field and suppose that $I$ is an ideal of $F[x]$ which contains an irreducible polynomial of degree $n$ and a nonzero polynomial of degree less than $n$. Show that $I=F[x]$. I can't ...
1
vote
3answers
381 views

Each element is a square of some element

I have to show that each element of $\mathbb{F}_{2^n}$ is a square of some element. Could you give me some hints how I could do that??
2
votes
1answer
45 views

Acyclic chain complex and contracting chain homotopy

Let $R$ be a Ring and $(C_k, d_k)_{k\geq0}$ a acyclic chain complex of free modules, meaning $im(d_{k+1})=\ker(d_k)$ for all $k$. I want to show that there is a family of R-module-homomorphisms ...