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
31 views

Is my understanding of this corollary correct?

The following is a theorem/corollary pair in an introductory abstract algebra course. Theorem: $f(x)\equiv g(x) $ mod $p(x)$ if and only if $[f(x)]=[g(x)]$, where $[h(x)]=h(x)$ mod $p(x)$. ...
5
votes
3answers
32 views

Possible textbook redundancy concerning invertible mappings

In my textbook (Modern Algebra by John Durbin, 6th Ed), there is the following theorem: Let $S$ denote any nonempty set. (a) Composition is an associative operation on $M(S)$, with identity ...
-1
votes
2answers
61 views

polynomials root finding [on hold]

Is every root of a polynomial of positive integer degree n, and with a rational coefficients is considered algebraic number? and how one can find some roots to this polynomial ...
1
vote
1answer
26 views

Homomorphism on U(36)

Question Suppose that $f$ is homomorphism of $U(36)$, $\ker(f) = \{1,13,25\}$, and $f(5) =17$. Determine all the elements that map to 17. What I've tried so far So I've determined that $U(36) = ...
3
votes
1answer
46 views

Modular Forms: Find a set of representatives for the cusps of $\Gamma_0(4)$

We write $SL_2(\mathbb{Z})$ as $\coprod\alpha_i\Gamma_0(4)$, where $\coprod$ means the disjoint union, and the $\alpha_i$ are the coset representatives of $SL_2(\mathbb{Z})\diagup\Gamma_0(4)$. We ...
-2
votes
0answers
72 views

Show a well-defined morphism [on hold]

Question: Let $L⊆A^{*}$ be a language recognised by a monoid $M$ via a morphism $\theta:A^{*}\rightarrow M$ such that $\theta$ is onto. Show that $\phi:M\rightarrow M(L)$ defined by $\phi(\theta(w)) ...
-3
votes
0answers
34 views

Let $F =\mathbb Z_3$. Give an example of irreducible monic polynomial in F[x]. [on hold]

Let $F =\mathbb Z_3$. Give an example of irreducible monic polynomial in F[x]. Then give an extension field $\mathbb Z_3 < H$ and classify according to the fundamental theorem of abelian groups. ...
1
vote
1answer
23 views

question about isomorphism involving Dihedral group.

suppose $D_{n}$ is dihedral group with order $2n$, do we have this Isomorphism below? $$D_{2k+1} \times \mathbb{Z}_2 \simeq D_{4k+2}$$ I think it is wrong, I couldn't find any mapping, but I ...
0
votes
0answers
66 views

Infinite Non Abelian 3- Group [on hold]

Does there exist a infinite non Abelian Group whose every non identity element has order 3.
9
votes
3answers
138 views

What do groups and rings “look like”?

Taking undergraduate physics courses, I had to deal with Euclidean vectors often. In classes like Calc III, the concept was also there. I'm not sure if this is why, but I've always had a more ...
1
vote
3answers
51 views

Finite field definition question

My textbook says that $ (\mathbb{Z}_{m},+,*)$ is a field if and only if m is a prime number. However, on Wikipedia it says: "Finite fields only exist when the order (size) is a prime power $p^{k}$ ...
3
votes
1answer
49 views

Show that $\mathbb{C} \otimes_\mathbb{Z} \mathbb{C} \cong \mathbb{C} \otimes_\mathbb{Q} \mathbb{C}$

Show that $\mathbb{C} \otimes_\mathbb{Z} \mathbb{C} \cong \mathbb{C} \otimes_\mathbb{Q} \mathbb{C}$ This is not homework, it is part of an answer of Show that $\mathbb{A}_\mathbb{C}^2 \ncong ...
1
vote
1answer
22 views

Find the Existence and Uniqueness of a Cyclic subgroup

Giving that $|H|$ is cyclic. If $|H|=n$ then for each $a>0$ such that $a|n$ there exist a unique subgroup of $H$ of order $a$. This subgroup is Cyclic subgroup $\langle x^d\rangle$ where $d= ...
2
votes
2answers
57 views

How do i proof that that the map $ \varphi: Aut(F_n) \to GL_n(\mathbb{Z})$ is homomorpism?

I'm trying to proof that a map $ \varphi: Aut(F_n) \to GL_n(\mathbb{Z})$ is a homomorphism but i can't exactly define which is the function to show that. The map $ \varphi $ is a map with for any ...
0
votes
0answers
17 views

Whether a given collection is a set or not [duplicate]

I originally knew that a set is a concept that has no definition. However, today, in abstract algebra class, the professor told us that the collection of all fields E such that E/F is an alegbraic ...
1
vote
1answer
19 views

Does the restriction of scalars functor preserve the quotient module construction?

I have two rings $R,\,S$ (not necessarily commutative - in the case I have in mind, just $R$ is commutative) such that $i:R\subset S$. Clearly, if we have a module $M$ over $S$, then we can restrict ...
0
votes
1answer
19 views

Compositum of normal extensions is a normal extension

I'm trying to prove that if $ F \subset K, F \subset M $ are normal extensions, $ K,M \subset E $, then $ KM$ is also a normal extension of $ F $. I tried using the fact that $ F \subset KM $ is a ...
1
vote
0answers
18 views

If $I$ is a finitely generated semisimple left $R-$module, show that $I=Re$ for an idempotent $e \in I$.

Let $I$ be a left ideal in a semiprime ring $R$. If $I$ is a finitely generated semisimple left $R-$module, show that $I=Re$ for an idempotent $e \in I$.
1
vote
1answer
45 views

Number Theory/Quadratic Number Rings

Show that if $u,v,x,y$ are positive integers for which $u^2+2v^2=x^2+2y^2=p$ a prime number, then $u=x$ and $v=y$. I get that if we had $\alpha=u+v\sqrt{-2} \in \mathbb{Z}[\sqrt{-2}]$, then the ...
2
votes
1answer
99 views

Irreducible Polynomials over Finite Fields [on hold]

How would I show that $p(x)=x^5+x^2+1$ is an irreducible polynomial over $\Bbb Z_2=\{0,1\}$.
3
votes
1answer
34 views

simple question on conjugacy classes

if $ \;G = \langle a,b\;|\; a^9 = b^3 = 1, bab^{-1} = a^4\rangle\; $ of order $\;27\;$ Then how would i show that $b$ is conjugate to $ba^3$ I have been fiddling around with this for ages and cannot ...
2
votes
1answer
54 views

Class group of $\mathbb Q(\sqrt{-55})$ and finding representatives for ideal classes

My first step in computing the class group of $\mathbb Q(\sqrt{-55})$ was to compute the Minkowski bound. Initially, I said $\lambda(-55)=2\sqrt{-55}/\pi<2(8)/3<6$ and I went the normal way of ...
0
votes
1answer
33 views

Where does the term “affine space” come from?

I'm wondering since few years what its origin is. The adjective affinis means neighbouring, allied to, kindred and the noun derived from it affinitas means relationship, connection, union, affinity. ...
4
votes
2answers
47 views

Finding the character table of this group

if $ G = <a,b| a^9 = b^3 = 1, bab^{-1} = a^4> $ of order 27 then know the following, that any element can be written as $b^ka^n$ with n $\in [0,8], k\in[0,2]$ and that the 11 conjugacy classes ...
0
votes
3answers
42 views

Minimal polynomial of $\sqrt{2} + \sqrt{3}$ over $\Bbb{Q}(\sqrt{6})$

I have to find the minimal polynomial of $\alpha = \sqrt{2} + \sqrt{3}$ over $\Bbb{Q}(\sqrt6)$. $\alpha^{2} = 2 + 2\sqrt6 + 3$ so $f(X) = X^{2} - 5 - 2\sqrt6$ is a polynomial where $f(X) \in ...
0
votes
2answers
32 views

Degree of extension

How to find the degree of $\mathbb Q\left(\sqrt2+\sqrt[3]2\right) $ over $\mathbb Q\left(\sqrt2\right)$ ? I know how to find $\mathbb Q\left(\sqrt2\right)$ over $\mathbb Q$. But i am confused in ...
10
votes
2answers
294 views

Is algebra needed to really understand and/or enjoy model theory?

What are the desirable pre-requisites to be able to learn model theory well? In particular, it seems that connections to algebra are used heavily especially as examples. I would like to know if a ...
0
votes
0answers
10 views

center of s-unital rings

Is there a right s-unital ring whose center is zero? One noncommutatve algebra that came to my mind it was the ring of infinite matrices over a field, but this algebras has nonzero center. If ...
1
vote
0answers
20 views

Isomorphic encryption or homomorphic encryption?

Many encryption functions are said to be homomorphic: http://en.wikipedia.org/wiki/Homomorphic_encryption As encryption functions are invertible, they can be considered one-to-one and onto on ...
1
vote
1answer
33 views

Finding the dimension of $\mathbb{F}[x]/p(x)$

Let $\mathbb{F}$ be a field, let $p(x) \in \mathbb{F}[x]$ and let $\text{deg}(p) \geq 1$. I'm looking at a proof which shows that $\mathbb{F}[x]/p(x)$ is an integral domain implies that ...
2
votes
2answers
47 views

Is the minimal number of generators of an ideal the rank of the ideal as a free $\mathbb Z$-module?

In an algebraic number theory course, my lecturer said that any ideal of $\mathcal O_K$, where $K$ is a quadratic number field, is generated by at most two elements. I am wondering why this is. When ...
2
votes
1answer
22 views

Determine the Galois group of $ F(x^5) \subset F(x) $

I'm rather new to Galois theory and have been given this exercise: Suppose $ F $ is respectively equal to $ \mathbb{Q}, \mathbb{C}, \mathbb{F}_5 $ (the third one is just the 5-element field). My task ...
3
votes
1answer
80 views

Question about ideals of a ring: $I\cdot J=I \implies J=I$?

Doing exercises, this question came to my mind. Is it true that if $I$ and $J$ are proper and nonzero ideals of a ring $R$, $$I\cdot J=I \implies J=I?$$ And $$I\cdot J=I \iff J\subseteq I?$$
1
vote
1answer
39 views

What are the possible sets of critical values of a complex polynomial of degree $n$?

Question: What are the possible sets of critical values of a degree $n \ge 2$ polynomial with complex coefficients? ($z\in\mathbb{C}$ is a critical value of $f$ if there is $w\in\mathbb{C}$ with ...
2
votes
0answers
37 views

Projective curve, local ring, maximal ideal, dimension $k+1$. [on hold]

Let $C \subset \mathbb{P}_2$ be a projective curve, and $p \in C$ a point of multiplicity $m$. If $\mathcal{O}_p(C)$ is the local ring of $C$ at $p$, and $\mathfrak{m} \subset \mathcal{O}_p(C)$ is its ...
1
vote
2answers
51 views

irreducible characters of a group

I am currently attempting a past exam paper and am stuck on the following question for part a) $\mu$ is an irreducible character iff it is equal to the character of an irreducible representation, ...
17
votes
2answers
242 views
+50

Without using Heegner-Stark-Baker, $\mathbb{Q}(\sqrt{-11})$ has class number $1$.

Prove that $\mathbb{Q}(\sqrt{-11})$ is of class number $1$. I have found that the ideal $(2)$ of the integer ring $\mathbb{Z}[(1 + \sqrt{-11})/2]$ of $\mathbb{Q}(\sqrt{-11})$ is a prime ideal. ...
2
votes
1answer
27 views

Direct decompositions and quotients of abelian groups

Let $G = \langle a \rangle_{27} \oplus \langle b \rangle_{81}$. Find a direct decomposition $G = \langle 10a + 60b \rangle \oplus ?$. Find the elementary divisors of $G/ \langle 3a + 18b \rangle$. ...
-1
votes
0answers
84 views

Properties of a localization of $\mathbb Z$

Let $R=\{\frac ab ∈ ℚ ∣ b \text{ is odd}\}$. (1) Prove that $R$ is isomorphic to $ℤ_P$, where $ℤ_P$ is the localization at $P$, for a prime ideal $P$ of $R$. (ii) Find $U(R)$. Prove that ...
0
votes
2answers
60 views

Ideals of the localization of a ring [on hold]

Let $R$ be a domain. Let $P$ be a prime ideal of $R$. (i) Prove that $S:=R\setminus P$ is a multiplicatively closed system with no zero divisors. Prove that $RP=S^{−1}R=\{a/b∈K\mid b∉P\}$. (ii) We ...
-2
votes
1answer
47 views

Ring morphism is injective or not [duplicate]

I've got an exercise here that states the following: Let $R$ be a commutative ring and $S$ be a subset of $R$, with $S$ multiplicatively closed. If $T$ is a commutative ring and $φ'\colon R \to T$ ...
1
vote
1answer
70 views

character tables and solubility

I am currently going through a past exam paper for a group theory module and am unable to answer the following section of a question. The copy of my lecture notes doesn't seem to have a section on ...
2
votes
1answer
20 views

Describing the Kernel of an identity-preserving ring morphism from a ring $R$ to an Endomorphism ring of an additive Abelian Group.

I'm currently working through TS Blyth's book on Module Theory (Module Theory: An Approach to Linear Algebra). From Exercise 2.3: "Let $M$ be an Abelian Additive Group, and $R$ a unitary ring. Let ...
8
votes
8answers
228 views

An example of a mapping $\eta\colon\mathbb{N}\to\mathbb{N}$ such that $\eta(x)=n$ has infinitely many solutions for each $n\in\mathbb{N}$

Suppose I have the mapping $\eta\colon\mathbb{N}\to\mathbb{N}$ such that for each $n\in\mathbb{N}$ the equation $\eta(x)=n$ has infinitely many solutions. I saw this question which is basically the ...
0
votes
1answer
76 views

Herstein Problem No.7 Page 102

let $G$ be a group of order $30$ .How many non-isomorphic groups of order $30 $ are there? Before doing this I have shown that every Sylow 3 and Sylow 5 subgroup is normal in G and G has a normal ...
5
votes
2answers
84 views

Proving Irreducibility of $x^4-16x^3+20x^2+12$ in $\mathbb Q[x]$

Trying to prove that the following polynomial is irreducible in $\mathbb Q[x]$: $x^4-16x^3+20x^2+12$ What I have tried: 1.) Eisenstein's Criterion, but there exists no suitable prime. 2.) ...
0
votes
0answers
17 views

Polynomial with Complex Coefficients Find largest value [closed]

Suppose $f(x)$ and $g(x)$ are coprime monic polynomials. Now suppose that both have complex coefficients. Now, $f(x)\rvert g(x)^2-x$ and $g(x)\rvert f(x)^2-x$. $\forall f$ with degree of $100$, how ...
1
vote
2answers
64 views

Problem about ideals of the localization of a ring

I'm having problems on doing the section (ii) of this exercise. Let $R$ be a domain. Let $P$ be a prime ideal of $R$. (i) Prove that $S:=R\setminus P$ is a multiplicatively closed system with no ...
2
votes
1answer
18 views

Let $d$ be a non-square discriminant. If $I \subset \mathcal{O}_d$ is a nonzero ideal, then $I \cong \mathbb{Z}^2$ as an abelian group.

For every integer $d \equiv 0,1 \mod 4,$ we define a quadratic ring $\mathcal{O}_d$ by: If $d \equiv 0 \mod 4,$ let $\mathcal{O}_d = \mathbb{Z}\left[\sqrt{\dfrac{d}{4}} \right].$ If $d$ is a ...
1
vote
0answers
29 views

Summary of Galois quintic unsolvability

Is this a sufficient flow of logic? Summary: Consider a fifth degree polynomial that has a formulaic solution. Then we have a radical extension with a string of subgroups of the Galois group which ...