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.

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2
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2answers
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There is no nontrivial ring homomorphism between two commutative rings with unity and characteristic of distinct primes

The following is an old exam question and the question is: Show that there is no nontrivial ring homomorphism between two commutative rings with identity if their characteristics are distinct primes. ...
7
votes
3answers
135 views

How can I prove irreducibility of polynomial over a finite field?

I want to prove what $x^{10} +x^3+1$ is irreducible over a field $\mathbb F_{2}$ and $x^5$ + $x^4 +x^3 + x^2 +x -1$ is reducible over $\mathbb F_{3}$. As far as I know Eisenstein criteria won't ...
1
vote
1answer
37 views

$M_1$, $M_2$ and $M_1\cap M_2$ injective imply $M_1+M_2$ is also injective?

Let $M$ be an $R$-module ($R$ is a ring with identity) and let $M_1$ and $M_2$ be two injective submodules such that $M_1\cap M_2$ is also injective. How to show $M_1+M_2$ is injective? If the ...
1
vote
1answer
54 views

Learning Galois theory - required subtopics that are prerequisite?

This is not a reference request, that is, I have access to many textbooks I am happy with. What I don't know is, what are the things I need to know to get started? My idea on the path of knowledge ...
0
votes
0answers
21 views

Fiber product of $f$ and $g$ is isomorphic to $\mathbb Z\oplus \mathbb Z_p$?

Let $p$ be a prime number. I'm supposed to show the fiber product (pullback) of the canonical projections $f:\mathbb Z\longrightarrow \mathbb Z_{p}$ and $g:\mathbb Z_{p^2}\longrightarrow \mathbb Z_p$ ...
0
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0answers
21 views

Proper formulation of one-to-one and onto proofs for group isomorphism

I have to construct an isomorphism for the two groups. I have the isomorphism itself but I'm not sure if my formulation is correct in regard to proving the mapping being 1-1 and onto and I don't want ...
8
votes
1answer
71 views

$A$ regular, $k'/k$ transcendental. How to prove that $A \otimes_k k'$ is regular?

Let $k$ be a field and $k'$ a purely transcendental extension of $k$. Let now $A$ be an integral finitely generated $k$-algebra. How to prove that if $A$ is regular then $A \otimes_k k'$ is also ...
0
votes
1answer
38 views

Question about an inverse limit.

Define a partial order on $\Bbb{N}$ to be $n \leq m$ iff $n = m $ or $n |m$ and there's a twin prime dividing $m$ and not $n$. It's easy to see that it's a poset. Define a system of abelian groups ...
3
votes
2answers
74 views

The subring of $k[x, y]$ generated by $\{x y^{i}: i\geq 0\}$ is not finitely generated over $k$ [duplicate]

Let $R$ be the subring of $k[x, y]$ generated by $\{x y^{i}: i\geq 0\}$. Can someone explain why $R$ is not finitely generated as a ring over $k$ (i.e. finitely generated as a $k$-algebra)? By ...
1
vote
1answer
48 views

Prime ideals in $R[x]$, $R$ a PID

Let $R$ be a PID. Show that if $r \in R$ and $$p = (r, \underline{f}(x), \underline{g}(x))$$ is prime, where $\underline{f}(x), \underline{g}(x) \in R[x]$ are nonconstant irreducible polynomials, ...
5
votes
2answers
78 views

Prime ideal $P$ of $\mathbb{Z}[x]$ such that $P \cap \mathbb{Z}=\{0\}$ is principal

The problem stated more precisely is this: Let $P$ be a prime ideal of $\mathbb{Z}[x]$ such that $P \cap \mathbb{Z} =\{0\}$. Show that $P$ is a principal ideal. I think there is a problem with my ...
2
votes
0answers
49 views

When a two-generated ideal of a noetherian integral domain have a finite projective resolution?

Let $R$ be a noetherian integral domain, and $I$ a non-zero ideal of $R$ which can be generated by two elements. (We do not know if $I$, considered as an $R$-module, is $R$-projective; maybe yes maybe ...
2
votes
1answer
36 views

Conjugates of Sylow $p$-groups in $GL_3(F_p)$

In this list of review questions, there is the following question about $GL_3(F_p)$. Question 1.38. Let $G$ be the group of invertible 3 × 3 matrices over $F_p$, for $p$ prime. What does basic ...
3
votes
3answers
59 views

Irreducible polynomial over $\mathbb{Q}(\zeta)$

Show that the polynomial $f(x)=x^5-2$ is irreducible over $\mathbb{Q}(\zeta)$, where $\zeta=e^{2\pi i/5}$. I tried show that the roots of polynomial $f(x)=x^5-2$, $$\sqrt[5]{2}, \zeta\sqrt[5]{2}, ...
1
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0answers
49 views

Projectivity of a (prime) ideal in a noetherian integral domain

Assume $R$ is a noetherian integral domain (and assume $R \neq k[x_1,\ldots,x_n]$), $I$ is a non-zero ideal of $R$ ($I$ is finitely generated, since $R$ is noetherian), and $I$ is not necessarily ...
-1
votes
0answers
45 views

Prove $a_1a_2$ is commutative in ring $R$ if $a_1=a_2$ [closed]

I was attempting to prove that any element of a ring squared is commutative in any given ring, and didn't know where to begin with this.
0
votes
2answers
65 views

Create a field from set of 2 elements.

Can we always create a field from a set of at least $2$ elements? For addition I considered a function: $A\times A \rightarrow A$. If $a+b=b+a=a+a \rightarrow a $. If $b+b \rightarrow b$. Is it ...
6
votes
1answer
115 views

A game from Exercise in Artin's Algebra (Chapter 2 M.13)

I found an interesting problem in Chapter 2 for Artin's Algebra (2nd Ed) in the Miscellaneous section that I haven't been able to figure out. The text of the problem is quoted below. M.13 (a ...
-1
votes
3answers
62 views

Noncommutative algebraic operation. [closed]

Can we always find a non-commutative algebraic operation in a non-empty set?
1
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1answer
30 views

Example of a Non-Graded Ideal in a Graded Ring

A ring $S$ is said to be graded if there are additive subgroups $S_0, S_1, S_2, \ldots$ such that $S=\bigoplus_{k\geq 0}S_k$ and $S_iS_j\subseteq S_{i+j}$ for all $i$ and $j$. An ideal $I$ in a ...
3
votes
4answers
50 views

If $R$ is a ring with identity and $a$ is a unit, prove that the equation $ax=b$ has a unique solution in $R$.

So, this was my initial proof: Assume $R$ is a ring, and $a,b\in R$ Let $x_1$ and $x_2$ be solutions of $ax=b$ Hence, $ax_1=b=ax_2 \Rightarrow ax_1-ax_2=0_R \Rightarrow a(x_1-x_2)=0_R$ Thus, we ...
3
votes
0answers
63 views

Prove that $|\operatorname{Aut}(G)| \leq \prod_{i=0}^{k}(n-2^i)$ [closed]

Suppose that $G$ is a group of order $n > 1$, prove that $$|\operatorname{Aut}(G)| \leq \prod_{i=0}^{k}(n-2^i)$$ where $k=\lfloor \log_2 (n-1)\rfloor$.
2
votes
1answer
28 views

Derived subgroup of $S_n$ and $D_n$

I know that Derived/Commutator subgroup of $S_3$ is $A_3$ and commutator subgroup of $D_4$ is cyclic of order $2$. But What about derived groups of $S_n$ and $D_n$? How can I calculate them?
0
votes
2answers
37 views

Showing that any field extension of a finite field is simple

We know that the multiplicative subgroup $F^\times$ of a finite field $F$ is cyclic. Use this to show that any field extension of a finite field is simple. Any clues?
2
votes
2answers
70 views

If $R$ is a commutative ring, the nilpotents form necessarily an ideal of $R$? [duplicate]

This is an algebra question from an exam a few years ago: Let $R$ be a ring, and let $N = \{a \in R: a^n = 0 \text{ for some } n \in \mathbb{N}, (n \text{ depends on } a) \}$. Prove or disprove: ...
-1
votes
0answers
26 views

Direct sum of algebras [closed]

I was wondering if we have an algebra $$A=F_p[h_1, …, h_n] \oplus F_p[h_1, …, h_n] g$$ for all $p$, can we have the direct sum over Z? Thanks
0
votes
1answer
19 views

What is a relation (finitely related module)?

https://en.wikipedia.org/wiki/Finitely-generated_module#Finitely_presented.2C_finitely_related.2C_and_coherent_modules I've understood the first part of the definition. Then, "M is isomorphic to ...
-1
votes
1answer
19 views

I have to show it is isomorphic to $K = GF(p^{kd})$ [closed]

Suppose $F = GF(p^k)$ is a finite field. I know $F[C]$ is a field extension of $F$ with degree $d = \deg m$, and I have to show it is isomorphic to $K = GF(p^{kd})$ (where $C$ is a companion matrix ...
2
votes
2answers
54 views

Can one decompose any ring into a (possibly infinite) product of indecomposable rings?

Let $R$ be a ring. Do there exist indecomposable rings $R_i$, $i\in I$, such that $R={\prod}_{i\in I} R_i$?
9
votes
1answer
201 views
+50

Unsolvability of a Quintic and its link with “Simplicity” of $A_{5}$

At the outset I must mention that I don't have a fairly working knowledge of Galois Theory (but do have some idea of group theory in the sense that I can understand normal subgroups). I read the ...
2
votes
0answers
28 views

Intermediate “prime” extensions [Rotman]

Problem Assume $F$ contains the $k$th roots of unity, and let $R=F(\alpha)$, where $\alpha$ is a root of $x^k-a$ for some $a\in F$. Prove that there exist intermediate fields $$F=K_0\subset ...
2
votes
1answer
39 views

Global dimension of an intermediate ring

Assume $A \subseteq B \subseteq C$ are noetherian integral domains, where $A$ and $C$ have the same finite global dimension $n$. Also assume that $C$ is a finitely generated $B$-algebra and $B$ is a ...
25
votes
1answer
310 views

Group $G$ whose center $Z(G)$ is cyclic and with $G/Z(G)$ commutative

I have some issue to solve following exercise. The exercise is from a French book on Algebra (cours d'Algèbre) from Jean Querré. The book is from the 1970's. If the center $Z(G)$ of a group $G$ is ...
0
votes
1answer
28 views

Number of Cosets of Intersection of Subgroups

Similar question has been asked on SE before but the problem statement is usually more specific and gives more information (in particular, tells you what to prove), but this problem asks to prove or ...
2
votes
1answer
84 views

Is a nontrivial finite group of order $n$ always isomorphic to a subgroup of $GL_{n-1}(\mathbb{Z})$?

I saw this question on an old qualifying exam: Let $G$ be a group of order $n\ge2$. Is such a group always isomorphic to a subgroup of $GL_{n-1}(\mathbb{Z})$? A simpler problem would be to show ...
3
votes
1answer
57 views

Isomorphic or equal?

Let $\sim_n$ be the usual equivalence relation of congruence modulo $n$ in $\mathbb{Z}$, i.e., for $a,b\in\mathbb{Z}$, $a\sim_nb\Leftrightarrow a-b=k\cdot n$ for some $k\in\mathbb{Z}$. For $n=0$ the ...
4
votes
3answers
92 views

Can you learn linear algebra with an abstract algebra book?

I am trying to learn both linear and abstract algebra. I already took a basic Matrix Theory course using Anton's linear algebra book (not very rigorous) and I want to self study the rest of the ...
0
votes
0answers
16 views

Number of homomorphisms between finitely generated abelian group and a finite cyclic group

This is the situation: Suppose we have a finitely generated group $G= \mathbb{Z}^r \times E$ with $E$ it's tortion subgroup and $m=$ exponent of $E$ i.e. the least natural number such that $x^m=1$ ...
0
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0answers
38 views

Prime ideal in indecomposable commutative ring [closed]

Let $R$ be a commutative indecomposable ring with Jacobson radical $J$. When can we find a prime ideal contained in $J$?
3
votes
1answer
45 views

Difference between $(N+P)/N$ and $P/N$

If $N$ and $P$ are submodules of the $A$-module $M$ (where $A$ is a commutative ring with unity), why is there a difference between $(N+P)/N$ and $P/N$? If $x\in (N+P)/N$ then $x=n+p+N=p+N$ for some ...
2
votes
1answer
41 views

Why is $x^{100} = 1 \mod 1000$ if $x < 1000$ and $\gcd (x,1000) = 1$?

Let $U(1000) =$ the multiplicative group of all integers less than and relative prime to $1000$. "Show that for every $x \in U(1000)$ it is true that $x^{100} = 1 \mod 1000$." Been thinking ...
0
votes
0answers
17 views

construction field $F_n$ [duplicate]

I know that if $F_n$ is a finite field, then $n$ should be a prime power. I want some sources in order to learn how I can construct finite fields $F_n$ for such an $n$.
2
votes
1answer
39 views

Need a hint to get started on this algebra problem.

The problem is: If it is known that a group G with m generators and exponent n is finite, can one conclude that the order of G is bounded by some constant depending only on m and n? Equivalently, are ...
0
votes
2answers
39 views

Prove that $M$ is finitely generated and it is a semi-simple module.

Let $M$ be a $R$-module. It is given that intersection of all maximal sub-modules of $M$ is the zero module. Moreover the module is given to be Artinian. Prove that $M$ is finitely generated ...
2
votes
2answers
84 views

Classifying groups of order $5 \cdot 11 \cdot 61$

My question is whether I'm classifying groups of order $5 \cdot 11 \cdot 61$ correctly. (This is a qualifying exam question, so I also want to make sure that I'm doing it “efficiently”.) Sylow's ...
2
votes
3answers
38 views

Prove that if $M$ is finitely generated then it is Artinian.

Let $M $ be a semisimple $R$-module. Prove that if $M$ is finitely generated then it is Artinian. To show this we have to prove that every non-empty collection of sub-modules of $M$ has a minimal ...
2
votes
1answer
27 views

is $\mathbb{Q} ^{|\mathbb{R}|} ‎\times‎ \{1 \}$ divisible subgroup of $ \mathbb{Q} ^{|\mathbb{R}|} ‎\times‎ \mathbb{Z}_2$?

According to Unit Groups of Classical Rings by Karpilovsky, p.107 we know that: If $F$ is a real-closed field, then $F^*‎\simeq‎ \mathbb{Q} ^{|F|} ‎\times‎ \mathbb{Z}_2$. Now, we know that ...
3
votes
1answer
42 views

A ring isomorphic to its square [duplicate]

Is there an example of a ring $A$ (with unity) which is isomorphic as unital rings to $A\times A$? Any such ring can't have invariant basis number so in particular can't be commutative.
1
vote
1answer
31 views

Trouble with a proof: $(p^n - 1 , e)=1$ for $e\in \mathbb{N}$, p prime

I'm having trouble understanding a proof. The Lemma states: For every natural number $e$ there are infinitely many prime powers $q$ with $(q-1,e)=1$. The prove is as follows: Write $e=2^km$, m odd. ...
0
votes
0answers
49 views

Transitivity of discriminant for flat algebras

Let $A$ be an finite flat $R$-algebra and $A'$ be an finite flat $A$-algebra such that it is also finite flat as an $R$-algebra. Then we have a notion of discriminant ideals ...