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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Irreducible but not prime element

I am looking for a ring element which is irreducible but not prime. So necessarily the ring can't be a PID. My idea was to consider $R=K[x,y]$ and $x+y\in R$. This is irreducible because in any ...
0
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1answer
28 views

Proving Fermat's Little Theorem with Lagrange

I know how to prove Fermat's little theorem using the binomial expansion and induction. How can I prove it using Lagrange's theorem? So I want to show $c^p\equiv c\pmod p$, i.e. $c^{p-1}\equiv 1\...
9
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1answer
62 views

Finite subgroup of $\text{SO}(3)$ acts on set of points on unit sphere in $\mathbb{R}^3$ which are fixed via some nontrivial rotation in $G$

Let $G$ be a finite nontrivial subgroup of $\text{SO}(3)$. Let $X$ be the set of points on the unit sphere in $\mathbb{R}^3$ which are fixed by some nontrivial rotation in $G$. I have two questions. ...
1
vote
1answer
22 views

Why $\mathbf Z[\sqrt{-5}]/(2, 1+\sqrt{-5})\simeq \mathbf Z[x]/(2,x+1,x^2+5)$?

Why is $(2, 1+\sqrt{-5})$ not principal? \begin{align*}\mathbf Z[\sqrt{-5}]/(2, 1+\sqrt{-5})&\simeq \mathbf Z[x]/(2,x+1,x^2+5)\simeq \mathbf Z_2[x]/(x+1,x^2+1)\\ &=\mathbf Z_2[x]/\bigl(x+1,(...
0
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0answers
17 views

Simultaneous row echelon form

I am working on the following problem right now: Let $V,W,H$ be finite dimensional vector spaces. I have a group action $Gl(V)\times Gl(W) \curvearrowright Hom(V\otimes H,W)$ in the obvious way i.e. ...
1
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0answers
17 views

$|G|=p^kq$, $p,q$ prime, $p^k\leq 2q$ implies one of the Sylow subgroups is normal

I could use some help with this one: Let $G$ be a group, $|G|=p^kq$ where $p,q$ are prime and $p^k\leq 2q$. Prove that at least one of the Sylow subgroups is normal.
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0answers
41 views

Is there a field extension $K / \Bbb Q$ such that $\text{Aut}_{\Bbb Q}(K) \cong \Bbb Z$?

I'm not requiring this extension to be Galois, that's why I wrote $\text{Aut}$ instead of $\text{Gal}$. I'm not very familiar with infinite extensions nor with profinite groups. I don't know if my ...
3
votes
1answer
29 views

dimension of subspace $V = \{A \in M_{2\times2}(\mathbb{C}) : \textrm{tr}(A) = 0, A^\textrm{T} = -\bar{A}\}$

Let V be the subspace defined by $V = \{A\in M_{2\times2}(\mathbb{C}) : \textrm{tr}(A) = 0, A^\textrm{T} = -\bar{A}\}$ and note $V$ is a vector space over $\mathbb{R}$ with inner product $<A,B> ...
4
votes
1answer
91 views

When is a given polynomial a square of another polynomial?

I meet a problem in which I hope to show a special polynomial is not a square of another polynomial. More precisely, let's consider the polynomial $$f(x):= 1-x+2bx^n-2bx^{n+1}-b^2x^{2n-1}+2b^2x^{2n}-b^...
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+50

Regarding the general method of the ''Classify groups of order $X$'' question.

Anyone who has had to prepare for an algebra qualifying exam is familiar with the "Classify groups of order $X$" question. To illustrate my general question, which I postpone until the end, consider ...
2
votes
1answer
42 views

Is $4x^2+3xy^2+y^3+7$ irreducible in $\mathbb{C}[x,y]$?

Is $4x^2+3xy^2+y^3+7$ irreducible in $\mathbb{C}[x,y]$? I tried to group it like $(4)x^2+(3y^2)x+(y^3+7)$. This is a polynomial with degree $2$ so I am thinking of applying quadratic formula... where ...
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0answers
20 views

What's an example of a non-normal, purely inseparable field extension?

Certainly every purely inseparable simple extension must be normal, since the minimal polynomial of the generating element $a$ must look like $X^{p^r}-a$, which splits in the extension, so the example ...
0
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0answers
21 views

Different Representation Matrices from same Generating Set

Motivation: This post. $K \subset S_n, \langle K \rangle =G \leq S_n$. We can create a Representation Matrix $M$ from $K$ that represnts $G$ (Furst. Hopcroft, Luks). Question: Is $M$ unique for a ...
7
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0answers
51 views

If $A[X] \cong B[X]$ as rings, are the degrees of irreducible polynomials the same in $A$ and in $B$?

First, I ask my question and then I add some explanations: Suppose that $A$ and $B$ are two commutative rings such that $A[X] \cong B[X]$ as rings. Denote by $D_A$ the set of all positive integers ...
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0answers
53 views

Clifford algebra of a non-diagonal quadratic form over rings

I know how to construct explicitly the clifford algebra of a quadratic form over fields, even in the case the diagonal quadratic form over rings. But how should I $\textbf{construct explicitly}$ the ...
4
votes
2answers
171 views

Why is $S/Z$ a domain for the ideal $Z=\{z\in S\mid za=0,\;\forall a\in R\}$ in $S$?

Suppose $R$ is a rng with no zero divisors, not necessarily commutative. I know $R$ can be embedded into a ring $S:=\mathbb{Z}\times R$ by identifying $r\in R$ with $(0,r)\in S$. The operations on $S$ ...
2
votes
1answer
30 views

Norm of Gaussian integers: solutions to $N(a) = k$ for $k \in \mathbb{N}$ and $a$ a Guassian integer

I have two questions. Which integers are equal to the norm of some Gaussian integer? In general, how many solutions does$$\text{N}(a) = k$$have for a given $k \in \mathbb{Z}$? I am investigating the ...
4
votes
2answers
61 views

What algebra is generated by $\mathrm{O}(2)$?

The unit complex numbers can be identified with the $2 \times 2$ special orthogonal matrices $\mathrm{SO}(2)$. The problem with $\mathrm{SO}(2)$, however, is that its not closed under $\mathbb{R}$-...
4
votes
1answer
49 views

Quotient ring of Gaussian integers $\mathbb{Z}[i]/(a+bi)$ when $a$ and $b$ are NOT coprime

The isomorphism $\mathbb{Z}[i]/(a+bi) \cong \Bbb Z/(a^2+b^2)\Bbb Z$ is well-known, when the integers $a$ and $b$ are coprime. But what happens when they are not coprime, say $(a,b)=d>1$? — For ...
15
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1answer
110 views

Does $A^2 \cong B^2$ imply $A \cong B$ for rings?

If $A$ and $B$ are two unital rings such that $A \times A \cong B \times B$, as rings, does it follows that $A$ and $B$ are isomorphic (as rings)? I believe that the answer is no, but I can't come ...
0
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0answers
38 views

$\{g\in G\mid\alpha(g)=g^{-1}\}=\frac34|G|$, find an abelian subgroup of index 2

$G$ is a finite group, $\alpha$ is an automorphism of $G$ and $I=\{g\in G\mid\alpha(g)=g^{-1}\}$. If $|I|=\frac34|G|$, show that $G$ has an abelian subgroup of index 2. Related question I don't ...
0
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2answers
27 views

Question about proof of finding Aut$(\Bbb Z_{10})$

The question I have is from one line in this proof: To begin with, observe that once we know $\alpha(1)$, we know $\alpha(k)$ for any $k$, because $\alpha(k) = k\alpha(1)$. So, we need only ...
0
votes
1answer
21 views

Property of free modules

Question is regarding following property of free modules: Let $P$ be a free $R$ module. To every surjective homomorphism $f:B\rightarrow C$ of $R$ modules and to every homomorphism $g:P\...
3
votes
1answer
114 views

Why is ring of integers $\mathcal O_K$ called ring of integers - what properties of $\mathbb{Z}$ does it inherit?

I was wondering why ring of integers $\mathcal O_K$ for field $K$ is called ring of integers. Definition says that elements in this ring will be a solution for monic equation with coefficients of ...
0
votes
1answer
29 views

projection of plane on a line

In basic algebra by Nathan Jacobson, he said that a plane can be projected to a line. The text states that "if the plane is the domain and a line is co domain,then one maps any point P in the plane on ...
5
votes
1answer
41 views

Does there always exist an irreducible polynomial of degree $d$ over $\mathbb{Z}/p\mathbb{Z}$? [duplicate]

Let $p$ be a prime and let $d$ be a positive integer. Does there always exist an irreducible (i.e. unfactorable) polynomial of degree $d$ over $\mathbb{Z}/p\mathbb{Z}$?
0
votes
2answers
44 views

What does Aut$(\Bbb Z)$ look like?

What does Aut$(\Bbb Z)$ look like? (Integers with the operation of addition) I understand that it's the set of all automorphisms from $\Bbb Z$ to $\Bbb Z$, or Aut$(\Bbb Z) = \{\alpha_1, \alpha_2, ... ...
0
votes
2answers
25 views

Help with proving a fact about injective modules

Let $R$ be a ring with identity. I am trying to show that an $R$-module $A$ is injective if and and only if for every left ideal $L$ of $R$ and every $R$-module homomorphism $g: L \rightarrow A$, ...
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0answers
65 views

Mathematics Colleges [on hold]

Can anyone please give me the names of US colleges with the best mathematics program for undergraduates? I came to know about Carneige Mellon university, but it does not offer financial aid to ...
5
votes
0answers
59 views

If $|G|=p^3q^2$ then $\Phi(G)$ is cyclic for primes $p\neq q$.

I have conjectured this result for the Frattini subgroup by doing some calculations in GAP. I think this is even true if $|G|=p_1^{i_1}\cdots p_n^{i_n}$ for $i_j\leq 3$ holds, but I would like to ...
0
votes
1answer
27 views

A question on extension of rings which related to their direct summands

I read "Foundations of Module and Ring Theory" of Robert Wisbauer and I got stuck in this problem: *Show for a ring $R$. The following assertions are equivalent: (a) $R$ has a unit. (b) If $R$ is ...
0
votes
1answer
23 views

possible cyclic group from fundamental theorem of finite abelian

Question: Give a representative of each Isomorphism class of Abelian group of order 225. Which ones are cyclic? By the Fundamental theorem of finite abelian group: $\left | G \right |=225=3^{2}...
2
votes
1answer
32 views

What does additive mean in “additive basis” in algebraic geometry?

Let $Gr(k,n)$ be the set of all $k$-dimensional subspaces of an $n$-dimensional vector space. I heard that people say "an additive basis" of $\mathbb{C}[Gr(k,n)]$. What does additive mean? Thank you ...
22
votes
1answer
239 views

How smooth can non-nice associative operations on the reals be?

Suppose ${*}:\mathbb R\times\mathbb R\to\mathbb R$ is $\mathcal C^k$ and associative. Does it necessarily satisfy the identity $a * b * c * d = a * c * b * d$? For $k=0$ the answer is "no" -- a ...
1
vote
3answers
48 views

Does the converse of Lagrange's theorem hold for any finite Dedekind group?

I'm currently reading a book on algebra and the following argument is used to prove that the converse of Lagrange's theorem (if $d$ divides $|G|$ there exists a subgroup of order $d$) holds for any ...
1
vote
3answers
63 views

Is my proof True ? ( about Group theory, direct product )

I have a problem. It states that: Let $G$ is a group and $|G|=mn$, $(m,n)=1$. Assume that $G$ has exactly just one subgroup $M$ with order $m$ and one subgroup $N$ with order $n$. Prove: $G$ is ...
3
votes
1answer
38 views

$R$ be a Noetherian domain , $t\in R$ be a non-zero , non-unit element , then is it true that $\cap_{n \ge 1} t^nR=\{0\}$?

Let $R$ be a Noetherian domain, $t\in R$ be a non-zero, non-unit element, then is it true that $$\bigcap_{n \ge 1} t^nR=\{0\} \text{?} $$ It almost feels like the nilradical (which is zero for any ...
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1answer
81 views

Is this regular function globally rational?

Let $\mathbb{k}$ be an algebraically closed field of characteristic not 2 or 3, and let $X \subseteq \mathbb{A}^2_\mathbb{k}$ be the locally closed subset given by $X = \{ (x,y) : x^3=y^2, (x,y) \...
1
vote
2answers
103 views

Nilpotent or non-Nilpotent Jacobson Radical

Let $R$ be a ring with identity element such that every ideal of which is idempotent or nilpotent. Is it true that the Jacobson radical $J(R)$ of $R$ is nilpotent? If $R$ is Noetherian and $J(R)$ is ...
3
votes
2answers
80 views

Question in finding a new $\mathbb{Q}$-basis for $F/\mathbb{Q}$.

Let $F$ be the splitting field of $x^4 - 2$ over $\mathbb{Q}$. Let $G$ be its Galois group. When viewed as a $\mathbb{Q}$- vectorspace, $F$ has the following basis: $$\mathcal{B}=\{1,2^{1/4},2^{1/2},...
2
votes
1answer
79 views

For which category (if any) are Lie algebras the algebras of a monad?

I was reading about monads recently, and it came to me that the purpose of the category of algebras of a monad seems to be to switch to a "representation" which is easier for computations. Soon after ...
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votes
2answers
46 views

How to prove a given subset is a subgroup [on hold]

If $A, B$ are additive subgroups of a ring $(R,+,\cdot)$, then prove that the set $AB=\{r\in R:r=\sum_{i=1}^n a_i b_i \textrm{ for }a_i\in A,b_i\in B\}$ is an additive subgroup of $R$
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0answers
33 views

Question about generators and Hom functor

In a given category $\mathcal{C}$ I want to prove the following statement: If $U$ es a generator in the category $\mathcal{C}$ if and only if the left-exact functor $Hom_{\mathcal{C}}(U,-): \mathcal{...
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vote
3answers
40 views

Subgroup that generates $\mathbb{Z}$

For reference, the example in question is taken from Contemporary Abstract Algebra (Gallian): $$\left< 8, 13 \right> = \mathbb{Z}$$ My first question is to confirm that this is saying that $...
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votes
0answers
21 views

How can I prove that the the number of elements of order $k$ in $\mathbb{Z}_n$ is φ(k)? [on hold]

How can I prove that the number of elements of order $k$ in $\mathbb{Z}_n$ is ϕ(k) where $k$ is number that divides n ?
3
votes
4answers
60 views

If $G$ is a non-abelian group of order 10, prove that $G$ has five elements of order 2.

I'm trying to prove this statement: If $G$ is a non-abelian group of order $10$, prove that $G$ has five elements of order $2$. I know that if $a\in G$ such that $a\neq e$, then as a ...
18
votes
4answers
618 views

Problems from the Kourovka Notebook that undergraduate students can fully appreciate

The Kourovka Notebook is a collection of open problems in Group Theory. My question is: could you point out some (a "big-list" of) problems [by referencing them] presented in this book that are, ...
3
votes
3answers
112 views

Infinitely many primes $\equiv 3 \mod 4$

Question 1 Is the following proof of the infinitude of primes $\equiv 1\mod 4$ okay? Consider a prime divisor $p\mid (n!)^2+1$. Then $(n!)^2\equiv -1 \mod p$, hence $n!$ has multiplicative order $4$ ...
5
votes
0answers
53 views

Is an algebraic field extension $k \subseteq K$ normal if $\mathrm{Aut}_{K}(\bar{k})$ is a normal subgroup of $\mathrm{Aut}_{k}(\bar{k})$?

Over a perfect field $k$ it is well known that an algebraic field extension $k \subseteq K$ is normal if and only if $\mathrm{Aut}_{K}(\bar{k})$ is a normal subgroup of $\mathrm{Aut}_{k}(\bar{k})$, as ...
0
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0answers
39 views

Deciphering the theorem of perfect powers

Can someone help me understand what happened to this equation from the paper entitled Perfect Powers With All Equal digits but one...I don't understand the part when it lets a and c not equal to zero....