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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Class-equation of $\mathbb Z_2$ $\oplus$ $S_3$.

$S_3=${${(1),(12),(13),(23),(123),(132)}$}. $\mathbb Z_2=${${0,1}$}. $\mathbb Z_2$$\oplus$$S_3$={(0,(1)),(0,(12)),(0,(13)),(0,(23)),(0,(123)),(0,(132)),(1,(1)),(1,(12)),(1,(13)),(1,(23)),(1,(123)),(...
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1answer
25 views

Isomorphic quotient rings of polynomial rings

I wish to know if $$\dfrac{\mathbb Z_2[x_1,x_2,x_3,x_4]}{\langle x_1x_3,x_2x_4,x_1+x_3,x_2+x_3+x_4\rangle}\cong\dfrac{\mathbb Z_2[a,b]}{\langle a^3,b^2,a^2-ab\rangle}.$$ Context - I was computing the ...
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5answers
60 views

Why are an even number of flips required to get back to the original list?

Consider the list of numbers $[1, \cdots, n]$ for some positive integer $n$. Two distinct elements $i$ and $j$ of the list can be switched in a so-called flip. For example, let $f$ be a flip that ...
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1answer
40 views

Let $R$ be a ring with unity. Consider $X^2 - 1$ $\in$ $R[X]$. Then $X^2 - 1$ has at most two roots in R.

Indicate True/False Let $R$ be a ring with unity. Consider $X^2 - 1$ $\in$ $R[X]$. Then $X^2 - 1$ has at most two roots in R. I need a hint to solve this problem. I have tried some common rings ...
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1answer
16 views

Criterion of separability of polynomials

When I study page 270 in Lang's algebra, I have a problem. Let $f(X)=X^3+aX+b$ be an irreducible polynomial over a field $k$. Lang says that if char $k$ is not equal to $2,3$, then $f$ is separable. ...
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1answer
48 views

What makes an algebra finite?

What makes an arbitrary algebra finite? If an algebra $A$ is generated by an infinite set of generators, but the operations of $A$ are finitary, is $A$ finite or infinite? If $A$ is generated by a ...
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35 views

On oblath's theorem [on hold]

it is just my first encounter about this topic ,it is the topic that my prof gave to me in my undergrad studies.I found it interesting but there are still parts(like theorem) in this topic which make ...
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2answers
40 views

Operation of permutations on functions

Let $P$ be the additive group of mappings from $\mathbf{Z}^n$ to $\mathbf{Z}$. For $f \in P$ and $\sigma \in \mathfrak{S}_n$ (the symmetric group of degree $n$) let $\sigma f$ be the element of $P$ ...
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2answers
20 views

A simple group such that $[G:H]=n$ can be embedded into $A_n$

Let $G$ be a finite simple group and $H$ be a proper subgroup of $G$ such that $|G:H|=n$. Then, how do I prove that $G$ can be embeded into $A_n$? I can prove that $G$ can be embedded into $S_n$ ...
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2answers
40 views

Polynomial ring with arbitrarily many variables in ZF

For a given field $k$ and a set $X$ we want to define the ring $k[X]$ of polynomials with $X$ as the set of variables. We do not assume $X$ to be finite. And we want to do this without employing axiom ...
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2answers
42 views

Linking two theorems: on algebraic closure and on minimal splitting field

Consider following two statements from same book of Cohn: Basic Algebra. Proposition 7.3.2: If $k$ is any field and $\mathcal{F}$ is a set of polynomials over $k$, then any two minimal splitting ...
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18 views

Techniques to turn expressions involving integer roots into polynomials by substitution?

Inspired by this question involving an Equation for a Torus How to find a parametrization for a torus? I started wondering if there is some systematic approach to do substitutions to make equations ...
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2answers
45 views

Does associativity imply closure?

Does associativity of binary operation imply closure under this operation? Sometimes definitions of semigroup, group or vector space omit axiom of closure under corresponding operations and sometimes ...
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1answer
45 views

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
23 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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43 views

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

There are 267 not isomorphic groups of order 64. How many of these are Abelian? Describe them.
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19 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
36 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
37 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 ...
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1answer
32 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
32 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
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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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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
52 views

Algebra, linear transformation, minimal polynomial [on hold]

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
49 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 ...
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1answer
70 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 $T_n v = 0$. ...
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1answer
89 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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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
51 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
71 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 ...
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1answer
53 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
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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
75 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 $f(x) = a_nx^n+....+a_0 \in \mathbb{Q}[x]$ 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\leq ...
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1answer
38 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
24 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
42 views

Solve the nth zero of a function. [on hold]

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)? ...
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2answers
65 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
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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 $\...
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2answers
69 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
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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 $c \in F-\{0\}$ $2) \deg x^{n_1}...
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1answer
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$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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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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2answers
73 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 ...