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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Deciphering the main theorem of the paper ''On Oblath's Problem''

I am trying to read the paper On Oblath's Problem, and I'm have difficulty understanding the main theorem. I can read the theorem but I don't understand it. May someone help me to make this theorem as ...
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2answers
30 views

Modules as morphisms to endomorphism rings

An $A$-module $M$ may be thought of as a (surjective) ring homomorphism $f: A \to E(M)$, where $E(M)$ is a ring of group endomorphisms of $M$. Then $am = f(a)(m)$. Is there any more to this ...
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50 views

There are 267 not isomorphic groups of order 64. How many of these are Abelian? Describe them. [closed]

There are 267 not isomorphic groups of order 64. How many of these are Abelian? Describe them.
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2answers
25 views

Group order 168 having a normal subgroup of order 4, then G has a normal subgroup of order 28.

Prove that if G is a group of order 168 that has a normal subgroup of order 4, then G has a normal subgroup of order 28. resolution: P4 is the ordeme subgroup 4 and P7 the order subgroup 7. As P4 is ...
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1answer
39 views

Is every subgroup of $S_n$ the Galois group of some extension of $\mathbb{Q}$?

Is every subgroup of $S_n$ the Galois group of some extension of $\mathbb{Q}$? It is well known that most (in some suitable sense) polynomials $f \in \mathbb{Q}[x]$ of degree $d$ and coefficients $|...
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1answer
45 views

On the maximum number of Sylow subgroups

I have some doubts regarding the first sentences below the fourth case in the more elementary solution by Prof. Samuel to problem 11856 of the Monthly. Here you have a screenshot of the relevant part ...
2
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1answer
34 views

Showing that $A \otimes_B A \to A$ is a surjective homomorphism.

Let us define a homomorphism $\phi: A \otimes_B A \to A$ by $a \otimes a' \to aa'$ where $A$ is a $B$-algebra, and both $A$ and $B$ are commutative rings. I want to show that this is a surjective ...
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2answers
34 views

If a magma M is both a semigroup and a quasigroup, is it necessarily a group?

If a magma which has an identity element is also a semigroup and a quasigroup, it can be shown that this is indeed a group. I'm looking for a counter example: a magma which is a quasigroup (for every ...
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0answers
20 views

Find if an element of $(\mathbb{F}_{2^w})^l$ is invertible

Give an element $a(x) \in (\mathbb{F}_{2^w})^l$ (that I think is a vector space) modulo $l(x)=x^4+1$ with a ring structure because $l(x)$ is reducible, I like to know if this element is invertible and,...
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42 views

A problem with infinitely many eigenvalues on a finite dimensional vector space

I want to develop some theory before posing the problem. Kindly stay with me. Consider $ Aut (k[x_1,...,x_n])$ where $k$ is an algebraically closed field, you can take $k=\Bbb C$. $\alpha \in Aut(k[...
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2answers
59 views

Algebra, linear transformation, minimal polynomial [closed]

Let $T : M_{n×n}(\Bbb F) \to M_{n×n}(\Bbb F)$ the linear transformation defined by $T (A) = AB$, for some matrix $B \in M_{n×n}(\Bbb F)$ fixed. Show that the minimal polynomial of $T$ coincides with ...
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4answers
53 views

Meaning of $Gal(L/L)$ for some field $L$?

In my notes it says $Gal(L/L)=1$ and I am confused on the notation clearly there is only one automorphism of $L$ that map all elements of the base field $L$ to itself namely the identity map. But what ...
9
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1answer
84 views

Does $\forall v ( T_1 v = 0 \lor T_2 v = 0 \lor \dots \lor T_n v =0 )$ imply $T_1 = 0 \lor T_2 = 0 \lor \dots \lor T_n = 0$?

Let $V$ and $W$ be vector spaces and $T_1$, $T_2$, $\dots$, $T_n$ be linear transformations from $V$ to $W$, such that for every $v$ in $V$, either $T_1 v = 0$, $T_2 v = 0$, $\dots$ or $T_n v = 0$. ...
4
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1answer
97 views

The other $47$ roots of the minimal polynomial for $\cos 1 ^\circ$

The minimal polynomial for $x=\cos 1 ^\circ=\cos \frac{\pi}{180}$ is: $$281474976710656 x^{48}-3377699720527872 x^{46}+18999560927969280 x^{44}- \\ -66568831992070144 x^{42}+162828875980603392 x^{40}-...
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0answers
15 views

Auslander-Reiten theory: exercise $23.b$ of 'Elements of the Representation Theory of Associative Algebras'

I am solving exercise $23.b$ of chapter IV of 'Elements of the representation theory of associative algebras' by Assem, Simson and Skowronski. The question is the following: Consider the following ...
2
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1answer
64 views

Irreducible elements for a commutative ring that is not an integral domain

Why does the definition of an irreducible element require us to be in an integral domain? Why can we not define an irreducible element exactly the same in a commutative ring that is not an integral ...
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4answers
78 views

Showing that for $f \in K[x]$, we have $f(x) \mid f(x + f(x))$

Let $K$ be a field an $f \in K[x]$. I now want to show that $f(x) \mid f(x + f(x))$ (in $K[x]$). I know that I need to find a polynomial $g \in K[x]$ so that $f(x) g(x) = f(x + f(x))$. So I thought ...
3
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1answer
57 views

Why Differential Forms on Riemann surfaces?

I am working with Rick Miranda's "Algebraic Curves and Riemann Surfaces". Right now I am in chapter four "Integration on Riemann Surfaces" and struggle with it a lot!:( It starts with the definition ...
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0answers
31 views

Congruence numbers

Having read about Stirling numbers of the second kind I am curious. The article says it shows the number of equivalence relations on a set $n$ with $k$ equivalence classes which makes sense to me from ...
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2answers
86 views

How do I find the ideal $I+J$ and quotient $R/(I+J)$?

This is a homework problem: Consider the polynomial ring $R=\mathbb Z_2[x_0,x_1,\dots,x_n]$. Let $I=\langle x_0x_1\cdots x_n\rangle$ and $J=\langle x_0+x_1,x_0+x_2,\dots,x_0+x_n\rangle$. Find $I+J$...
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Galois group of a palindromic polynomial is not $S_n$?

Let $f(x) = a_nx^n+\cdots+a_0 \in \mathbb{Q}[x]$ be a palindromic polynomial; that is, the coefficients of $f$ satisfy $a_n = a_0$, $a_{n-1} = a_1$, and more generally $a_{n-i} = a_i$ for all $0\leq i\...
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1answer
44 views

Problems related to conjugacy classes and their sizes.

Is $Z( G_1 \oplus G_2 \oplus G_3 \oplus\cdots\oplus G_n) = Z(G_1) \oplus Z (G_2) \oplus Z(G_3) \oplus \cdots \oplus Z$ $(G_n)$ true, where $ G_1, G_2, G_3 \cdots G_n$ are finite groups?$Z$,here refers ...
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1answer
27 views

compact metric space definition by closed covers

My book says the following: A metric space is compact iff: $$M=\cup A_{\lambda}\implies M = A_{\lambda1}\cup\cdots\cup A_{\lambda_n}$$ where each $A_{\lambda}$ is open. Then, it says that if $A_\...
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0answers
45 views

Solve the nth zero of a function. [closed]

Say I have a mystery continuous function, could be anything. f(x) Assuming we don't know the distribution of the zeros of the function, Is there a known way to solve the nth zero (hits the x-axis)? ...
2
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2answers
76 views

A question about orthogonality

Let $\mathcal{A}$ be a unital $*$-algebra over $\mathbb{C}$ and let $a,b\in\mathcal{A}$ be projections, that is, $a=a^*=a^2$ and $b=b^*=b^2$. If $a+b=1$, then $ab=0$. This follows from - \begin{align*...
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Sum of divisors in $\mathbb{Z}_2[x]$

Let $A$ denote a polynomial on $\mathbb{Z}_2[x]$ and $\sigma(A)$ denote the sum of divisors on $A$. Let$$A = x^h(x + 1)^k P^l Q^{2n - 1},$$where $P$, $Q$ are irreducible polynomials with degree at ...
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1answer
23 views

Showing that $\mathrm{Rad}((0)) ≠ (0)$ implies $R^\times \subsetneq R[X]^\times$

Let $R$ be a commutative ring with $1$, and let $I ≤ R$ be an ideal. We call $\mathrm{Rad}(I) := \{r \in R: \exists n \in \mathbb{N}_0: r^n \in I\}$ the radical of $I$. I now want to show that if $\...
3
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2answers
95 views

What kind of $n^{th}$ order polynomials are solvable by a square matrix with integer entries?

Consider a polynomial (monic for simplicity): $$x^n+a_1x^{n-1}+\dots+a_{n-1}x+a_n=0$$ Here we assume the roots are complex numbers. $a_k$ are integers. Now consider the corresponding matrix ...
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5answers
60 views

Is $F[x,y]$ a Euclidean Domain?

I was wondering if this is just common knowledge. So far for a field $F$ and transcendental $x$ and $y$, I know one can define the degree by $1) \deg c =0$, for any $c \in F-\{0\}$ $2) \deg x^{n_1}...
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1answer
46 views

$k[x,y,z]/(y-x^2,z-x^3)\cong k[x]$, where $k$ is a field

This is generalizing from a previous question, which asks to prove that $k[x,y]/(y-x^2)\cong k[x]$. The way I proved that was by using the homomorphism $\phi:k[x,y]/(y-x^2)\to k[x]$, $\phi(\overline{f(...
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3answers
22 views

Misunderstanding the definition of a cycle (cyclic permutation)

Given a set $E$, let $\mathfrak{S}_E$ be the group of permutations of $E$. Definition.$\ \ $ Let $E$ be a finite set, $\zeta$ a permutation of $E$, and $\overline{\zeta}$ the subgroup of $\...
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0answers
37 views

Primitive solvable group

Let $G$ be a finite solvable group. Suppose that $G=HN$ for all minimal normal subgroups $N$ of $G$. To show that $H = G$ or $G$ is primitive If $N$ is a minimal subgroup of $G$ then $N$ is an ...
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1answer
82 views

Show that $R$ is a field

Let $R$ be a commutative ring with unit. If $R\neq 0$ such that each finitely generated $R$-module is free then $R$ is a field. In my notes there is the following proof: We need to show that ...
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0answers
15 views

Prove that integral closure of $\mathbb R[x,y]/(y^2-x^3-x^2)$ is $\left( \mathbb R[x,y]/(y^2-x^3-x^2) \right) \left[ \frac{y}{x} \right]$ [duplicate]

i have to give a proof of the Headline. I just showed, that $y/x$ is integral over $R:=\mathbb R[x,y]/(y^2-x^3-x^2)$. How do I show, that $\bar R = R[t]$ where $t=y/x$? Furthermore, I have to show, ...
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3answers
81 views

What is the meaning of $\Bbb{Q}[x]/f(x)$?

I am very confused with the meaning of $\Bbb{Q}(x)/f(x)$. Does it mean the set of all polynomials modulo $f(x)$? If it does then how can we say that $\Bbb{R}[x]/(x^2+1)$ is isomorphic to set of ...
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1answer
27 views

How can we find a generating set?

How can we find a generating set of the $\mathbb{Z}$-module $\mathbb{Z}$ that does not contain the basis which is $\{1\}$ ? I saw in my notes that such a set is the $\{2,3\}$. Why is this a ...
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1answer
52 views

Show that $\text{Hom}_R(R^n,M)\cong \prod_{i=1}^n\text{Hom}_R(R,M)$

Let $R$ be a commutative ring with unit and let $M$ be a $R$-module. It holds that $\text{Hom}_R(R^n,M)\cong \prod_{i=1}^n\text{Hom}_R(R,M)$, right? How could we prove this? Do we have to define ...
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0answers
25 views

Module over nonassociative algebra [closed]

I have $A$ an algebra $*$ that is commutative an have an identity $(a^2b)b=((ab)ba$ and asking me verify conditions conditions for a $A$-module $*$ $M$. How I would use Linearization of identity? $...
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1answer
25 views

In an $\Bbb{N}$-graded domain $A$, units are homogeneous

Let $A$ be a graded domain, with additive subgroups $A_n,\,\forall\,n\geq 0$, s.t. ${A_n\cdot A_m}\subseteq A_{n+m}\,\forall\,n,m\geq 0$, and $A=\bigoplus_{n=0}^\infty\, A_n$ as abelian groups. I wish ...
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0answers
29 views

Describe the prime elements of the ring $\mathbb Z[\sqrt{-2}]$ [duplicate]

I have a ring $\mathbb Z[\sqrt{-2}]$ and I need to describe all the prime numbers of that ring. How I can do that? Thank you
5
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2answers
123 views

Polynomials generating the same $p$-adic fields

I wonder if the following fact is true: Pick $l\in \mathbb N$ a number and let $f,g\in \mathbb Z_p[x]$ be monic polynomials with coefficients in the ring of $p$-adic integers such that $f\equiv g \...
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Injectivity of $R \to R[t]/(f)$ for non-constant $f\in R[t]$

Question: Let $R$ be a (unital commutative) ring and $f = a_0 t^n + \cdots + a_n \in R[t]$ a non-constant polynomial. What are (necessary and sufficient) conditions on the coefficients $a_0,\ldots,a_n ...
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2answers
38 views

Let $K $be a field and $f \in K[x]$. Then there exists a splitting field for $f$ over $K$

Let $K $ be a field and $f \in K[X]$. Then there exists a splitting field for $f$ over $K$. I don't understand what this means, I think I am interpreting it wrongly. Take $x^2+1 \in \Bbb{Q}[X]$ then ...
0
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1answer
37 views

Find the numbers at which the polynomial is irreducible over $\mathbb{Q}$

How can I find the integers $a$ at which the polynomial $f(x)$ is irreducible over the field $\mathbb{Q}$? Thank you! $$f(x) = 5x^4 - 6x^3 - ax^2 - 4x + 2$$
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3answers
57 views

Show that polynomial is irreducible over $\mathbb{Q}$

How I can prove that polynomial $f(x)$, where$$f(x) = x^4 + 3x^3 + 3x^2 - 5$$ is irreducible over $\mathbb{Q}$? Thank you
0
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1answer
58 views

Odd degree of extension field in Couveignes Square root method

I was reading the Couveignes method to find square root for Number Field Sieve(reference here page 4 first line). It says that for this method the degree of extension K/Q must be odd so that Norm(-x)=-...
1
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3answers
144 views

Does $K = \mathbb Q[X]/(X^4 - 2)$ contain the imaginary unit $i$?

Let $P(X) = X^4 - 2 \in \mathbb Q[X]$. a) Prove that $P(X)$ is irreducible. b) Prove that the field $K = \mathbb Q[X]/(P(X))$ is an algebraic extension of $\mathbb Q$ and find a generator of it....
0
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1answer
43 views

If for all $a\in A$, $a\neq 0$ exists $b \in A$ such that $ab\neq 0$ then prove that $A\cong T_A$.

$A$ is a commutative ring with identity, and $T_A =\{ T_a \mid a\in A\}$, where $T_a=ax$ for all $x\in A$. If for all $a\in A$, $a\neq 0$ exists $b \in A$ such that $ab\neq 0$ then prove that $A\...
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2answers
32 views

product of cyclic groups G1 and G2

Let $G_1= \langle a \rangle$, $G_2= \langle b \rangle$ be two cyclic groups of orders $m$ and $n$ s.t. $(m,n)>1$. Then which one is not true for the product group $G=G_1×G_2$? (a) $G$ is cyclic ...
1
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2answers
68 views

why are these two different in abstract algebra?

Let G be a nonempty set closed under an associative product,which in addition satisfies: (1) There exists an $e\in G$ such that $a.e=a \forall a \in G$ (2)Give $a \in G$, there exists an element $y(...