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)

3
votes
2answers
50 views

Is $\text{Ind}_{Z(G)}^G \rho$ irreducible or not for nonabelian group $G$?

Let $G$ be a nonabelian group with center $Z(G)$. Let $\rho: Z(G) \to \text{GL}_n({\bf C})$ be an irreducible representation. Is $\text{Ind}_{Z(G)}^G \rho$ irreducible or not?
2
votes
2answers
46 views

Labelling the Vertices of Dodecahedron

Dodecahedron has 20 vertices. I want to label them by $1,2,3,4,5$ with the following rule. The five vertices of each face should have different labels. Q. What ...
3
votes
2answers
39 views

Frobenius reciprocity, proof if $V$ irrep of $G$ then multiplicity of $V$ in regular rep of $G$ is $\dim(V)$

I know the standard proof of the fact that if $V$ is an irreducible representation of $G$ then the multiplicity of $V$ is the regular representation of $G$ is $\dim(V)$. Does there exist a proof using ...
4
votes
2answers
84 views

Finite cyclic subgroups of $GL_{2} (\mathbb{Z})$ [duplicate]

How could we prove that any element of $GL_2(\mathbb{Z})$ of finite order has order 1, 2, 3, 4, or 6? I am aware of the proof supplied here at this link: ...
7
votes
1answer
64 views

Assume that $(G, \ast)$ is a group and that every element $a \in G$ satisfies $a \ast a = 1$. Show that $(G, \ast)$ is abelian.

Assume that $(G, \ast)$ is a group and that every element $a \in G$ satisfies $a \ast a = 1$. Show that $(G, \ast)$ is abelian. To prove $(G,\ast)$ is abelian, we must show that it is ...
3
votes
1answer
112 views

A Finite Group of Nonprime Order which is Unique up to Cyclic Group

Maybe I have to wait until I learn and study more, but I just became curious. I know that every finite group of prime order is cyclic, and hence unique up to isomorphism. I have 2 questions about ...
0
votes
0answers
60 views

Solve $x^2 + 2 = y^3$ for integer $x$ and $y$ [duplicate]

I am asked to find all integers $x$ and $y$ which satisfy $x^2 + 2 = y^3$. I am given the hint that I should work in the unique factorization ring $\mathbb{Z}[\sqrt{-2}]$. So I could write the ...
0
votes
1answer
49 views

State the Fundamental Theorem of Algebra for polynomials with real coefficients

I am working through a set of problems given to me and I have the following two questions presented to me: State the Fundamental Theorem of Algebra for polynomials with complex coefficients. State ...
1
vote
1answer
41 views

Why is $\mathbb{Z}_2[X]$ a principal ideal domain?

I used that $\mathbb{Z}_2[X]$ is a principal ideal domain to understand something but than I realized that in the lecture, we only noted that if we have a field, its polynomial ring is a domain. How ...
0
votes
1answer
50 views

Why “even number of elements in Group” in this question is given?

I am trying to prove one question about group. "If finite group G has identity e and even number of elements, prove that there is "a" (not equal to "e") such that $a*a=e$." I just don't understand ...
0
votes
0answers
15 views

Proofs for PCA, LDA, ICA, HMM learning algorithms and other stuff

I was wondering if there is some kind of encyclopedia of website for all known math proofs. I'm more interested in statistics (PCA, ICA, LDA, Factor analysis, HMM learning, GMM learning) and algebra ...
0
votes
2answers
36 views

Show that $f(e_G) = e_H$

Assume that $(G, \ast)$ and $(H, \Box)$ are groups and that $f: (G, \ast) \to (H, \Box)$ is a homomorphism. Let $e_G$ and $e_H$ denote the identity elements of $G$ and $H$ respectively. Show that ...
1
vote
1answer
24 views

Field and vector spaces

I have the following questions that asks for a)dimension of $Q(i)$ as a Q-vector space and b)dimension of $C$ as an $R$-vectorspace Also is the dimension of $R$ as a $Q$-vector space finite? I think ...
0
votes
0answers
26 views

If $I=\langle 12 \rangle$, then $Rad(I)=\langle 6\rangle$

To show that if $I=\langle 12 \rangle$, then $Rad(I)=\langle 6\rangle$, I did the following: $$36=3 \cdot 12 \\ 6^2=36 \in I \Rightarrow 6 \in Rad(I) \Rightarrow \langle 6 \rangle \subseteq Rad(I)$$ ...
3
votes
1answer
53 views

Ext groups due to Yoneda: why is this class zero

Consider category of $\mathbb{K}[x]$ modules. Let $\mathbb{K}$ be trivial $\mathbb{K}[x]$ module i.e. $x$ acts by zero. Easy to see that $Ext^2 (\mathbb{K}, \mathbb{K}) = 0$. But there is exact ...
2
votes
2answers
55 views

Bijection between sets of ideals

Let $A$ be a ring and $\mathfrak{b}$ be an ideal of $A$. Prove that the assignment $$\mathfrak{c} \mapsto \mathfrak{c}/\mathfrak{b}$$ induces a one-to-one correspondence between the ideals of ...
1
vote
3answers
60 views

is complex number under absolute value a group?

I have just started going over abstract algebra. One of the question is $*$ is defined on $\mathbb C$ such that $a*b=|ab|$ I tried to check three axioms : 1) Associativity 2) identity 3) inverse ...
2
votes
1answer
45 views

$M$ is maximal, $P$ is prime but not maximal [closed]

If $R$ is commutative with $1 \in R$, then each maximal ideal of $R$ is also a prime. The reverse doesn't hold. For example, $R=K[x, y], P=\langle x \rangle, M=\langle x, y\rangle$. Then ...
0
votes
3answers
41 views

Paradox of Field & Integral Domain in Venn Diagram

Check out venn diagram from this link, which is the order of number systems that has been embedded in my mind since grade school. Notice here that the integer $\mathbb Z$ is "inside" the rational ...
11
votes
3answers
95 views

Prove that $2$, $3$, $1+ \sqrt{-5}$, and $1-\sqrt{-5}$ are irreducible in $\mathbb{Z}[\sqrt{-5}]$.

So the Norm for an element $\alpha = a + b\sqrt{-5}$ in $\mathbb{Z}[\sqrt{-5}]$ is defined as $N(\alpha) = a^2 + 5b^2$ and so i argue by contradiction assume there exists $\alpha$ such that $N(\alpha) ...
0
votes
1answer
49 views

Why 2*3=6 in finite field?

In a finite field $F$, say of order 7. Suppose I didn't known that it must isomorphic to $\mathbb{F}_7$ By the definition of field, there exists $1 \in F$, and hence exists $1+1$ (denoted by $2$), ...
0
votes
1answer
22 views

Congruence problem in the Euclidean domain $\Bbb Z[\zeta]$

Let $\zeta = \frac12 +\frac{\sqrt{3}}{2}i$. I've proven that $\Bbb Z[\zeta]$ is a Euclidean domain with the norm given by multiplication by the complex conjugate. I'd now like to solve the system of ...
1
vote
2answers
22 views

How to prove that N(u) = 1 if and only if u is a unit in $Z[\sqrt-5]$

The norm of an element $u=a+b\sqrt-5$ in $Z[\sqrt-5]$ is defined as $N(u)= a^2 +5b^2$, now if $N(u) = 1$ then $a^2+5b^2 = 1$ but then how would i prove that it's a unit !?
0
votes
1answer
28 views

Ring isomorphism and indempotent element

Let $R$ be a ring. How to show that $R\cong R_1\times R_2 $, where $R_1,R_2$ are nontrivial rings, if and only if there exist $e\in R,\ e\neq0,1$ such that $e^2=e$ ? I need only hints.
3
votes
0answers
64 views

(Soft Question) How active an area of research is Non-Commutative Geometry? [closed]

I am currently an undergraduate, but I am considering applying for a phd in algebraic geometry or a related field. I am quite interested in the link between non-commutative geometry and theoretical ...
3
votes
1answer
37 views

Isomorphism of rings (with parameters)

Let $p$ be a prime number. For each choice of $a,b\in \mathbb{F}_p,$ let $F(a, b)$ be the ring $\mathbb{F}_p[X]/(X^2+aX+b).$ Find all possible choices of $(a, b), (a', b')\in \mathbb F_p \times ...
2
votes
1answer
30 views

exists homomorphism of $G$-representations $\pi: V \to V$ with image $X$

Let $G$ be a group (not necessarily finite) and $F$ a field. Let $V$ be a $G$-representation. Suppose $V$ is isomorphic to a direct sum of irreducible representations. Let $X \subset V$ be any ...
2
votes
0answers
36 views

$K$ field, $f/g\in K(X)-K$, $f,g\in K[X]$ coprime then $[K(X):K(f/g)]=\max(\deg f, \deg g)$

1) Let $K$ be a field, $q=f/g\in K(X)-K$ with $f,g\in K[X]$ coprime. Show that $q$ is transcendent over $K$ and that $[K(X):K(q)]=\max(\deg f, \deg g)$. Calculate the minimal polyinomial of $X$ over ...
2
votes
1answer
37 views

Question on simple subgroup $H$ and a normal subgroup $N$, of $G$

This one is a bit strange to me, mainly the third hypothesis. It goes as follows: Given a group (finite) group $G$, and $N, H \leq G$ such that $N$ is normal in $G$, and $H$ is simple ...
0
votes
2answers
45 views

maximal ideals in $\mathbb{Z}_2[X]$

I am looking for maximal ideals in $\mathbb{Z}_2[X]$. I started by considering principal ideals. \begin{eqnarray*} \langle 0 \rangle &=& \{\}\\ \langle 1 \rangle &=& \mathbb{Z}_2[X] ...
-3
votes
2answers
37 views

Question regarding homomorphism in groups [closed]

Let $f:G\to H$ a group homomorphism and $\mbox{ker}f$ contains n elements. Prove that $\mbox{Im}f$ has either $n$ or $0$ elements.
2
votes
1answer
28 views

prime elements, irreducible elements, unique factorization (rings)

I am currently trying to understand different kinds of rings. Is my understanding of the following correct? Prime elements are always irreducible. The decomposition of a ring element into prime ...
4
votes
1answer
51 views

If $F$ is a free group then $g^2=h^2$ implies $g=h$ for $h,g\in F$

If $F$ is a free group then $g^2=h^2$ implies $g=h$ for $h,g\in F$. I've been trying to prove this given the definition of a free group $F$: given group $F$ and subset $X\subseteq F$, $F$ is free ...
1
vote
1answer
18 views

[Verification]$G$ is a group whereby $(a\cdot b)^{i} =a^i\cdot b^i$ for three consecutive integers $i$ for all $a, b \in G$, show $G$ is abelian.

If $G$ is a group in which $(a\cdot b)^{i} =a^i\cdot b^i$ for three consecutive integers $i$ for all $a, b \in G$, show that $G$ is abelian. Proof: Let $x$ be the smallest of the 3 consecutive ...
1
vote
1answer
44 views

Question about split monomorphisms of free modules over local rings

In May's notes on Cohen-Macaulay and Regular Local Rings, during the proof of Serre's theorem on page 9, he claims that if $R$ is a local ring and $\phi\colon F\to F'$ is a map of finitely ...
2
votes
2answers
49 views

Embed finite field in algebraic closed field

Let $\Omega$ be an algebraic closed field of characteristic $p$, then for any field $F$ of $q$ ($=p^a$) elements, $F$ can be embedded in $\Omega.$ I need above property to proof that every ...
1
vote
1answer
63 views

Galois extension of $\mathbb{Q}$ with Galois group $\mathbb{Z}/4\mathbb{Z}$ that contains $\mathbb{Q}(\sqrt{3})$?

Does there exist a normal extension $L ⊃ \mathbb{Q}(\sqrt3) ⊃ \mathbb{Q}$ with Galois group $\mathrm{Gal}(L/\mathbb{Q}) \cong \mathbb{Z}/4\mathbb{Z}$?
1
vote
2answers
25 views

Finding the order of all the elements in Group $\mathbb{Z}_{12}$

I know that the order of an element $a$ in a group $G$ is the smallest positive integer $m$ such that $a^m=e$ and so for $(\mathbb{Z}_{12},+)$ we have $[0]$ is the identity of order 1. $[1]$ is ...
9
votes
2answers
112 views

Prove that $\Bbb Z[\sqrt{2}, \sqrt{3}]$ does not equal $\Bbb Z[\sqrt{2} + \sqrt{3}]$.

Prove that $\Bbb Z[\sqrt{2}, \sqrt{3}]$ does not equal $\Bbb Z[\sqrt{2} + \sqrt{3}]$. This is homework. I want to prove that these are different sets. The first set is the smallest ring ...
1
vote
1answer
31 views

A question about opposite ring.

I am reading this article about opposite rings. Are there relevant (or important) results about those rings? What opposite rings for?
3
votes
2answers
46 views

Galois group, algebraic closure over maximal extension

Let $\overline{\mathbb{Q}}$ be the algebraic closure of $\mathbb{Q}$. Let $\alpha \in \overline{\mathbb{Q}}\setminus \mathbb{Q}$ and let $K \subset \overline{\mathbb{Q}}$ be a maximal extension of ...
5
votes
2answers
49 views

$n-1$ dimensional permutation module for $S_n$

Say $n \ge 5$. Let $P$ be the $(n-1)$ dimensional permutation module for $S_n$, i.e. the permutation representation on $\{(x_1, \dots, x_n) \in {\bf C}^n: \sum x_i = 0\}$. Prove that: $\wedge^2P$ ...
2
votes
2answers
71 views

Proving a Basis in a Field of Polynomials

Let $B=(f_0,...,f_n)$ be a sequence of polynomials in $\mathbb{F}[t]_n$ of degrees $0,...,n$, respectively. (i) Prove that $B$ is a basis of $\mathbb{F}[t]_n$. (ii) For $f\in\mathbb{F}[t]_n$, how to ...
2
votes
1answer
38 views

Direct Sum of Homology Groups and Connected Sums - Something's gone wrong.

I know that for any surfaces, $K, L$ forming $K$#$L$ (where # denotes the connected sum) is done by deleting a disk from each and gluing together the boundary. I also know a few facts about the ...
0
votes
1answer
23 views

Norms on a Euclidean domain

A norm $N$ on a Euclidean Domain $R$ is a function $N:R \longrightarrow \{0,1,2,...\}$ such that (i) $N(a) = 0 \longleftrightarrow a = 0$ (ii)$N(ab) = N(a)N(b)$ (iii) if $b \neq 0$, then $a=qb + ...
1
vote
1answer
49 views

Prove $G$ is cyclic if for every $n\in \Bbb{N}$, there are at most $n$ solutions to the equation $x^n=e$. [duplicate]

Let $G$ be a finite group. Prove G is cyclic if for every $n\in \Bbb{N}$, there are at most $n$ solutions to the equation $x^n=e$. I am stuck with this :( Would appreciate your help.
2
votes
1answer
41 views

Polynomial ring, prime ideal, factor ring

I want to prove that this ideal: $I=(y^3-xz, xy^2-z^2, x^2-yz)$ is prime in $K[x,y,z]$. I think it would be a good idea to prove that the factor ring $K[x,y,z]/I$ has no zero divisors. In this factor ...
-2
votes
0answers
30 views

THE INVERSE RANK - MODULES [closed]

How can I find the inverse rank of 2 ( Zp) if p ??
2
votes
6answers
60 views

Ring homomorphism $\mathbb{Z}/8\mathbb{Z} \to \mathbb{Z}/5\mathbb{Z}$ implies $1=0$

Let $f$ be an ring homomorphism from $R_1$ to $R_2$ and define $f^*$ as the homomorphism from the group of units of $R_1$ to the group of units of $R_2$. Suppose $f^*$ is surjective, the question is ...
2
votes
1answer
34 views

finite/algebraic field extensions and minimal polynomial

1) To show: $L/K$ field extension is algebraic iff every subring with $K\subset R\subset L$ is a field. My answer: I can write $R=\bigcup\limits_{\alpha\in R}K[\alpha]$ with $K[\alpha]$ (field as ...