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)

4
votes
1answer
36 views

Extension $\mathbb{Q}(\sqrt{-5}, i)/\mathbb{Q}(\sqrt{-5})$, splitting.

In the extension $\mathbb{Q}(\sqrt{-5}, i)/\mathbb{Q}(\sqrt{-5})$, must principal prime ideals of $\mathbb{Z}[\sqrt{-5}]$ necessarily split into 2? Must nonprincipal prime ideals not split? ...
1
vote
1answer
42 views

How to prove that the ring of algebraic integers is a Bézout domain?

I was told that the ring of all algebraic integers (that is, the complex numbers which are roots of a monic polynomials with integral coefficients) is a Bézout domain. But I have no idea how to prove ...
0
votes
1answer
16 views

Why does it mean that $n$-th variable is removable?

I'm reading the proof for "the fundamental theorem of symmetric polynomials" and I have a trouble with it (http://en.m.wikipedia.org/wiki/Elementary_symmetric_polynomial) Let $P(X_1,...,X_n)$ be a ...
0
votes
3answers
24 views

$\gcd(a,n)\neq 1 \implies $ there is $b$ such that $ab\equiv 0 \pmod{n}$

I have that $\gcd(a,n)\neq 1$ ($a$ and $n$ are not coprime). Then, somehow, I need to prove that exists an $b$ such that $$ab\equiv 0 \pmod{n}$$ What I did: $$ab\equiv 0 \pmod{n}$$ is the same ...
3
votes
1answer
115 views

PID modulo a non-zero ideal is a semilocal ring

Let $R$ be a commutative ring, $\mathfrak{m}\subset R$ a maximal ideal and $f$ a monic polynomial in $R[x]$. I want to show that $A:=\frac{R[x]}{\mathfrak{m}[x]+(f)}$ is a semilocal ring, where ...
1
vote
2answers
40 views

Algebra and Maximal ideal.

I am trying to solve the following problem. If $ \mathcal{K}$ is a field and $a_1,a_2,\dots,a_n \in \mathcal{K}$. Prove that $(x_1-a_1,x_2-a_2,\dots,x_n-a_n)$ is a maximal ideal in ...
3
votes
1answer
62 views

Coker of powers of an endomorphism

Let $F\in\operatorname{End}_R(M)$, where $M$ is a Noetherian $R$-module. If $\operatorname{Coker}F$ is of finite length, is Coker and Ker of all powers of $F$ of finite length? Is the condition of ...
4
votes
2answers
68 views

Matrix ring $M_2(\mathbb{C})$, $\mathbb{C}^2$ with $M_2(\mathbb{C})$-module structure.

Let $R$ be the matrix ring $M_2(\mathbb{C})$. Let $M = \mathbb{C}^2$ with its natural $R$-module structure (just given by the usual action of $2 \times 2$ matrices on $2$-dimensional vectors). My ...
6
votes
2answers
66 views

The number of $3\times 3$ nilpotent matrices over $\mathbb{F}_q$ using the Orbit-Stabilizer theorem

The Fine-Herstein theorem says that the number of nilpotent $n\times n$ matrices over $\mathbb{F}_q$ is $q^{n^2-n}$. I am trying to verify this for the case $n=3$ using the orbit-stabilizer theorem. ...
2
votes
1answer
30 views

How to show the existence of an ideal in the ring of Gaussian integers that satisfy the following?

How to show that If $p$ is a prime and $p\equiv1\bmod4$ then there exists an ideal of the $R=\mathbb{Z}[i]$, the ring of Gaussian integers, such that $R/I$ is isomorphic to ...
0
votes
4answers
51 views

How do we find the minimal polynomial of $\alpha = a + b \sqrt{d}$ over $\mathbb{Q}$?

Let $\alpha = a + b \sqrt{d} \in \mathbb{Q} \left(\sqrt{d} \right) = \{a+b \sqrt{d}:a,b \in \mathbb{Q} \}.$ The minimal polynomial $m(x)$ of an algebraic number $\alpha \in \mathbb{C}$ is the monic ...
3
votes
1answer
44 views

Non-abelian Group with infinite exponent in which every proper subgroup has finite exponent

can you find a Non-abelian Group with infinite exponent in which every proper subgroup has finite exponent?
1
vote
0answers
22 views

Fundamental theorem of Algebra using ideas of complex singularities

Below is an excerpt from Arnold's Theory of Catastrophes (I haven't got an American edition, so translating from Russian). Where I can read about it in more detail? Not only regarding polynomials. ...
0
votes
1answer
13 views

For any monomial ordering, $1\leq m$ for any monomial $m$

Let $R$ be a ring. Let $\leq$ be a well-ordering on the set of (monic) monomials in $R[X_1,...,X_n]$. Then, $\leq$ is said to be a monomial ordering iff $mm_1\leq mm_2$ whenever $m_1\leq ...
6
votes
1answer
55 views

Looking for a non trivial homomorphism I

Is there a non trivial homomorphism $f: SU(2) \to O(2)$? Is there a concrete description of $Hom(SU(2), O(2))$?
1
vote
0answers
18 views
2
votes
2answers
35 views

A normal subgroup so that any homomorphism into a $p$-group is trivial on it.

Problem Let G be a finite group of order $n$ and $p|n$. Show that there is a unique normal subgroup $N$ satisfying the following property: (1)$G/N$ is a $p$-group (I guess it can be trivial group). ...
3
votes
2answers
64 views

Is the left translation $T_a(x) =ax $ a homomorphism?

I apologize if this is a super basic question but I was reading Lang's undergraduate algebra book and it says that the following function is a homomorphism: $$T_a(x) = ax$$ The way I would check if ...
1
vote
0answers
14 views

Is $\alpha$ a norm in the extension $K(\sqrt[n]{\alpha})$?

I'm having trouble wrapping my head around this. $K$ is a field of characteristic zero containing all $n$th roots of unity, and $\alpha \in K$. Let $L = K(\sqrt[n]{\alpha})$, $\mu$ the minimal ...
1
vote
0answers
9 views

Why is an $n$th power a norm in a Kummer extension?

Let $F$ be a $p$-adic field containing the $n$th roots of unity. Then by Kummer theory, $[F^{\ast} : F^{\ast n}]$ (which is finite) is equal to the cardinality of $\textrm{Gal}(E/F)$, where $E$ is ...
2
votes
1answer
41 views

Localising a polynomial ring and non-maximal prime ideal

I'm trying to work out the following past paper question and I've got stuck. $R$ is an integral domain and $S = R[t]$, the polynomial ring in one variable over $R$. We have that $Q$ is a prime ideal ...
2
votes
3answers
46 views

A finite group which has a unique subgroup of order $d$ for each $d\mid n$.

Problem Suppose G is a finite group of order $n$ which has a unique subgroup of order $d$ for each $d\mid n$. Prove that $G$ must be a cyclic group. My idea: I try to prove it by induction. Let ...
1
vote
1answer
50 views

Do non-graded rings exist?

I've been reading about graded rings recently, and I was wondering if there exist commutative or non-commutative rings for which no non-trivial gradings exist? That is, a ring for which if ...
1
vote
1answer
58 views

A morphism which is not a comorphism of a regular map

In the lecture, we dealt with morphisms, comorphisms and regular maps. The professor then brought the following example: Let $U$ and $V$ be quasi-affine sets over $\mathbb{C}$ and let $\psi \colon ...
17
votes
4answers
865 views

Why are ideals more important than subrings?

I have read that subgroups, subrings, submodules, etc. are substructures. But if you look at the definition of the Noetherian rings and Noetherian modules, Noetherian rings are defined with ideals ...
2
votes
1answer
64 views

Matrix and field extension

It is given that $F\subset K$ are fields. $A$ is a matrix of size $n\times n$ over $K$. I need to prove that there exist $c_1,\ldots,c_k\in K$, linearly independent over $F$, and matrices ...
3
votes
2answers
40 views

Algebra generated by a single element over an infinite field, does $A_1 \times \cdots \times A_r$ has the same property?

Let $K$ be an infinite field and $A_1, \ldots, A_r$ algebras over $K$ finite dimensional and such $\forall i = 1, \ldots, r \ \exists x \in A_i : A_i = K[x]$ (I think we say that $A$ is finitely ...
2
votes
1answer
48 views

Divisors of differentials.

Let $\mathbb{C}$ be the base field. Suppose $C \subset \mathbb{P}_2$ is a nonsingular projective curve of degree $d > 3$. Must it be the case that $D \in \text{Div}(C)$ is the divisor of a ...
3
votes
2answers
52 views

Proving that disjoint unions of presentations are coproducts of groups

I'm working through Aluffi's Algrebra: Chapter 0 and I need some assistance with an excercise. Aluffi, Ex. II.8.7 Let $(A|R)$, resp. $(A'|R')$, be a presentation for a group $G$ in Grp, resp. ...
1
vote
0answers
40 views

Show that if for $a \in \mathbb{Q}$ with $0 = f(a) $ for a monic polynomial $f(x)\in \mathbb{Z}[x] $, then $a \in \mathbb{Z}$

I. Show that if for $a \in \mathbb{Q}$ with $0 = f(a) \in \mathbb{Z}[x] $ for a monic polynomial, then in fact $a \in \mathbb{Z}.$ II. The set of algebraic integers of $\mathbb{Q}$ is ...
0
votes
1answer
20 views

Show that a representation of a finite group is isomorphic to its dual if its character takes only real values

This appeared as a part of showing that a representation of a finite group is isomorphic to its dual if and only if its character takes only real values. The "only if" part was easy to show. For the ...
3
votes
1answer
57 views

Is it possible to extend a commutative ring to have a unity? [duplicate]

Let $R$ be a commutative ring. Then, is it possible to extend this to have a unity? That is, is there a commutative ring with unity $R'$ such that $R$ is a subring of $R'$?
7
votes
2answers
90 views

For any rng $R$, can we attach a unity?

Let $R$ be an rng. (There may be no unity) Then, does there always exist a ring(with unity) $A$ such that $R$ is a subrng of $A$?
0
votes
1answer
38 views

Looking for a direct proof that all maximal ideals of $\mathbb C[x_1,x_2,…,x_n]$ are generated by $n$ linear polynomials

Without using Hilbert's Nullstelensatz , can we directly prove that all maximal ideals of $\mathbb C[x_1,x_2,...,x_n]$ is of the form $\langle x-a_1,x-a_2,...,x-a_n \rangle$ ? It is easy to prove it ...
7
votes
1answer
606 views

Proving that a polynomial is not solvable by radicals.

I'm trying to prove that the following polynomial is not solvable by radicals: $$p(x) = x^5 - 4x + 2 $$ First, by Eisenstein $p$ is irreducible. (It is not difficult to see that this ...
5
votes
2answers
146 views
+300

Structure of the group $\{1+p\mathbb Z_p \}$

In preparation for algebraic number theory I am reading Serre : A course in Arithmetic. I stuck in understanding a proof (p.17): Notation: $U_n=1+p^n\mathbb Z_p$ Actually there are many things ...
7
votes
5answers
2k views

Irreducible polynomial means no roots?

If a polynomial is irreducible in $R[x]$, where $R$ is a ring, means that it does not have a root in $R$, right? For example, to say that a polynomial $f(x)\in\mathbb Z[x]$ is irreducible in $\mathbb ...
2
votes
1answer
44 views

Any curve of genus three is either hyperelliptic or trigonal?

A curve $C$ is said to be trigonal if it admits a rational function $f: C \to \mathbb{CP}^1$ of degree $3$. Is any nonsingular plane curve of degree four trigonal? Can the map be chosen so that it has ...
1
vote
1answer
33 views

Proof that if $H_1 \leq G$ and $H_2 \leq G$ then $H_1 \cap H_2 \leq G$

I am trying to prove that Prove that if $H_1 \leq G$ and $H_2 \leq G$ then $H_1 \cap H_2 \leq G$ Unlike this question: Prove $H_1 \cap H_2 \le H_1 $ when $H_1, H_2 \le G$ and $H_1$, $H_2$ are ...
0
votes
1answer
48 views

What is orthogonal group $O(1)$?

I know that $O(2)$ is the group of 2x2 orthogonal matrices, but how can we extend the meaning of group and orthogonal to $O(1)$?
0
votes
2answers
54 views

Squares in $\mathbb Z_p$

Let $p\neq 2$, then I want to understand that every element in $\mathbb Z_p$ (p-adic integers) is a square. For the prove one must see that $2$ is invertible in $\mathbb Z_p$. But $2$ is the element ...
0
votes
1answer
41 views

Is there a function over $\mathbb{Z}_p$ that is never linear?

Let $p$ be a prime. I wonder if there is a function $f$ that satisfies the following rule: Whenever $$z_1 + \dots + z_c \equiv cx \mod p$$ (where $1 < c < p$, and $z_j, x \in \mathbb{Z}_p$) ...
0
votes
1answer
64 views

Prove the isomorphism of categories $Fun(\mathcal{A}\times\mathcal{B},\mathcal{C})\cong Fun(\mathcal{A},Fun(\mathcal{B},\mathcal{C})),$

I'm a computational engineer starting with a course of Introduction to Category Theory, and perhaps is extremely basic what I'm asking but I'm trying to learn how to make proofs in category theory ...
0
votes
1answer
23 views

Proving two matrices are cogredient over $\mathbb{Q}$

Two matrices $A,B$ are said to be cogredient if there exists an invertible matrix $P$ such that $B = P^{t}AP$. I know how to tell if two matrices are cogredient in algebraically closed fields, its as ...
0
votes
2answers
21 views

Show that if $\phi(F) \neq \{0\}$ then $F \cong R$.

Let $F$ be a field. Let $R$ be a ring and suppose $\phi : F \rightarrow R$ is an onto ring homomorphism. Show that if $\phi(F) \neq \{0\}$ then $F \cong R$. (Prove $F$ isomorphic to $F/\{0\}$ first) ...
9
votes
0answers
39 views

$M_1 \to M_2 \to M_3 \to 0$ exact iff $0 \to \text{Hom}_A(M_3, N) \to \text{Hom}_A(M_2, N) \to \text{Hom}_A(M_1, N)$ is exact.

Let $M_1 \to M_2 \to M_3 \to 0$ be a sequence of homomorphisms of $A$-modules. Is this sequence exact if and only if the induced sequence of abelian groups$$0 \to \text{Hom}_A(M_3, N) \to ...
3
votes
1answer
70 views

Characterize all finite unital rings with only zero divisors

Is it true that for every finite (for simplicity, commutative) ring $R$ in which every element not equal to $1$ is a zero divisor, is isomorphic to the zero ring or $\mathbb{Z}/2\mathbb{Z}$, ...
2
votes
1answer
30 views

Existence of multiplicative inverse in field $\mathbb{Q}(\sqrt{d})$

Let $\mathbb{Q}(\sqrt{d}) = \{a + b \sqrt{d}: a,b \in \mathbb{Q} \}.$ Show that $\mathbb{Q}(\sqrt{d})$ is a field. Everything seems obvious except for existence of inverses in the multiplicative ...
1
vote
0answers
34 views

If $G$ and $H$ are groups, prove that $(G \times H, x)$ is a group.

Prove that, if $(G,\ast)$ and $(H,\bullet)$ are groups, then the Cartesian Product $G \times H$ with the operation $(g_1,h_1) \circ (g_2, h_2) := (g_1 \ast g_2, h_1 \bullet h_2)$ $(G \times H, ...