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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Splitting fields and intermediates

If $F$ is a splitting field of $S$ over $K$ and $E$ is an intermediate field, then $F$ is a splitting field of $S$ over $E$.
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1answer
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$F$ need not be normal over $K$ [on hold]

Prove that if $F$ is normal over an intermediate field $E$ and $E$ is normal over $K$, then $F$ need not be normal over $K$. No clue.
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Collective name for algebraic structures

I am doing a thesis about various algebraic structures, primarely about groups, rings and modules (with maybe hint of algebras). However always having type out ALL of them constantly gets very tedious ...
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1answer
25 views

P.I.D. and a nontrivial ideal, Quotient ring has finitely many ideals [on hold]

A ring $R$ is a P.I.D. Let $I$ be a nontrivial ideal in $R$. Prove that $R/I$ has finitely many ideals.
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19 views

Classify all finite groups with property

Classify all finite groups $G$ with the following property: for every $H\vartriangleleft G$ there exists $K<G$ such that $G/H$ is isomorphic to $K$. My poor abstract-algebraic imagination doesn't ...
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28 views

defining gcd on rings

I see that in most textbooks they say let $R$ be an integral domain and start defining the greatest common divisor. My question is, can gcd's be defined on just commutative rings without an identity?
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How are the embeddings of a subfield of a Galois extension $K$ related to the embeddings of $K$?

Suppose we have a Galois extension $K/\mathbb{Q}$, then all embeddings of $K$ into $\mathbb{C}$ (or $\mathbb{R}$) are determined by the Galois group $G=\text{Gal}(K/\mathbb{Q})$. That is if we let ...
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3answers
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Inverse of $f: \mathbb{Z}_{21} \times \mathbb{Z}_{10} \to \mathbb{Z}_{210}$ such that $f(a,b)= 10a +21b$

Let $f: \mathbb{Z}_{21} \times \mathbb{Z}_{10} \to \mathbb{Z}_{210}$ such that $$f(a,b)= 10a +21b.$$ We have that $f$ is an isomorphism, but how does one go about finding explicitly the inverse ...
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Subgroup of order $9$ and $4$ in $\langle \alpha, \beta \rangle$

Let $$\alpha = (1,2,3,4,5)(6,7,8)(9,10,11)$$ $$\beta = (1,2,3)(4,5)(6,9,7,10,8,11)$$ We have that $\langle \alpha \rangle \cap \langle \beta \rangle = \{id\}$. So $$ord \langle \alpha, \beta \rangle ...
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1answer
10 views

Example of inverse semigroup with at least two idempotent elements

We say that the semigroup $S$ is inverse semigroup if for any $s\in S$ there is a unique $t\in S$ such that $sts=s,\ tst=t$. Suppose that $E(S)=\{e:\ e\in S,\ e^2=e\}$ and define $$s\sim ...
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1answer
40 views

A subset of a polynomial ring and its ideal. [duplicate]

Let $P=K[x_1,\dots,x_n]$ be a polynomial ring over a field $K$ and $I = (f)$ be a principal ideal in $P$ generated by $f \in P - \{0 \}$. Moreover let $L \subset \{x_1, \dots, x_n \}$ and $\hat{P} ...
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24 views

Help me with this Group Question

Let Sn := Bij({1, . . . , n}) the symmetric group with n elements equipped with composition of functions. (i) Show that Sn is a group. (ii) Let (e1, . . . , en) be the canonical base of Rn and Mn(R) ...
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39 views

Find the greatest common divisor of pairs of polynomials

I'm trying to find the greatest common divisor of $$p(x)=7x^3+6x^2-8x+4$$ and $$q(x)=x^3+x-2$$ where both $p(x),q(x)\in\mathbb{Q}[x].$ And if $d(x)=gcd(p(x),q(x)),$ I need to find two polynomials ...
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1answer
22 views

When does normal maximal subgroup have prime index?

Given finite group $G$, a normal maximal subgroup $H$, when is $[G:H]$ a prime? If $G$ is nilpotent, then the statement is true. But I am not sure about other $G$. Is there any counter-example ...
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17 views

$K$-affine $Hom$ functor relating to Polynomial Maps (Exercise in _Algebra: Chapter 0_ by Aluffi)

I'm currently learning some category theory and algebraic geometry, the basics at least. In Algebra: Chapter 0 by Aluffi, there is an exercise describing the relationship between morphisms of affine ...
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2answers
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$f\in L[x,y]$ such that $f(x,0)=0$ implies $f=y g$ with $g\in L[x,y]$?

Suppose $L$ is an infinite field (or even algebraically closed; I'm not sure if it is necessary to add that hypothesis). If we have a polynomial $f(x,y)\in L[x,y]$ and $f(x,0)\equiv 0$, does that ...
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Proving group and Morphisms of groups

Let Sn := Bij({1, . . . , n}) the symmetric group with n elements equipped with composition of functions. (i) Show that Sn is a group. (ii) Let (e1, . . . , en) be the canonical base of Rn and Mn(R) ...
3
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2answers
35 views

If $|F| = 8$ and $K$ is a subfield, show that $K = F$ or $K = \{0, 1\}$.

Let $K$ be a subring of a field $F$. If $|F| = 8$ and $K$ is a subfield, show that $K = F$ or $K = \{0, 1\}$. [Hint: Lagrange]. Lagrange's Theorem: If $H$ is a subgroup of a finite group $G$ I Then ...
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36 views

The center of a group

prove that for any group G, Z(G)=$\bigcap_{x\in G} C_{G}(\{x\})$ . In addition, show that if H$\subset$G , then $C_{G}(H)=\bigcap_{x\in H} C_{G}(\{x\})$ Z(G) is the center of a group $C_{G}(\{x\})$ ...
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If the tensor product of algebras $A \otimes B$ is unital, both $A$ and $B$ must be unital

It is clear that if $A$ and $B$ are unital algebras (over a field), then the tensor product $A \otimes B$ is also unital, with the unit being $1_A \otimes 1_B$. I came across an exercise that ...
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Kernel of a map in Graph Theory (toric ideals)

If we have an $n$-cycle with edges $e_1 =\{x_1,x_2 \}, e_2 = \{x_2, x_3 \},\dots, e_n = \{x_n,x_1\}$ with a $K-$algebra homomorphism $\phi: k[e_1,\dots, e_n] \to k[x_1,\dots, x_n]$ defined by ...
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Sylow counting to show group is isomorphic to semidirect product

Let G be a group of order $|G|=pq^m$ where $p, q$ are primes with $q^m<p$. Use a Sylow counting argument to show that $G\cong C_p \rtimes_h Q$ where $Q$ is a group with $|Q|=q^m$ and ...
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2answers
26 views

Show that a ring $R$ is a division ring if and only if, for each nonzero $a\in R$, there is a unique element $b\in R$ such that $aba = a$. [duplicate]

Show that a ring $R$ is a division ring if and only if, for each nonzero $a\in R$, there is a unique element $b\in R$ such that $aba = a$. $\Rightarrow$ Assume $R$ is a division ring. Let ...
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How to prove absence of a total order relation?

Show that on $\mathbb{C}$ (complex) there is no total order relation $≤$ such that both if the following properties hold $∀ (x, y, z) ∈ \mathbb{C}^3$, $x ≤ y \implies x + z ≤ y + z$ and $z ≥ 0, x ≤ ...
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How to show that a set of elements is a basis for the ring of integers of a number field?

Let $K$ be a number field of degree $n$ (that is $[K:\mathbb{Q}]=n$) with ring of integers $\mathcal{O}_K$. I know that there exists algorithms to find $\mathcal{O}_K$ and hence determine a ...
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1answer
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Sylow's theorem for group of $2$ by $2$ matrices of determinant $1$ over the field of order $3$

Let $G=SL(2,\mathbb{F_3})$ - group of $2$ by $2$ matrices of determinant $1$ over the field of order $3$. (a) Find the order of $G$. I think it is $24$ but not sure how to verify it. (b) ...
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1answer
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showing a function is surjective for isomorphisms

Consider a problem like the following. Prove that $ \mathbb{R} $ is isomorphic to the ring S of all $ 2 \times 2 $ matrices of the form $ \begin{bmatrix}a & 0\\0 & a\end{bmatrix}$, with $a ...
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29 views

Quick way to classify groups of small order

This is a past Qualifying exam on Algerbra : I'm curious if there is a quick way to solve Prob 1. To me, directly classifying groups takes too much time and I think I could not handle this in time ...
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Prove that these two fields are isomorphic.

I want to prove that $\bar{K}[V]/M_p \simeq \bar{K}$ where $K$ is a field, $\bar{K}$ is its algebraic closure and $$\bar{K}[V]=\bar{K}[x_1,...,x_n]/I_V,$$ where $I_V$ is the ideal attached to a ...
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Cosets of submodule

Suppose we have a ring $R$ and an ideal $J \subseteq R$. Let $M$ be an $R$-module and $N$ be a submodule of $M$. Assume for two elements $m$ and $m'$ of $M$ we have $$m+ JM=m'+JM.$$ Since $N ...
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1answer
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Representation of a group and its quotient

Let $G$ be a (finite) group and let $N$ be a normal subgroup of $G$. Suppose that we have a representation $(V,\rho)$ of $N$ and a representation of $(V, \tau)$ of the quotient group $G/N$. Here $V$ ...
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Let $G$ be a group of order $35$. Prove that $G \cong C_5 \times C_7$.

Let $G$ be a group of order $35$. Prove that $G \cong C_5 \times C_7$. $G$ has subgroups of orders $5$ and $7$ by Lagrange's theorem? If so, call them $A$ and $B$. I know their intersection is ...
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Order of group $G = \{A\in M_2(\mathbb{Z}_p): \mathrm{det}A= \pm 1 \}$

Also, $p>2$ is a prime number. Firstly, it's obvious that $G \leq GL_2(\mathbb{Z}_p)$, and we know that $|GL_2(\mathbb{Z}_p)|=p(p^2-1)(p-1)$. Next, we define the homomorphic map ...
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Mathematics and cinema

I wander if anyone of you have some knowledge about relations between abstract algebra and cinema. I'm not searching for movies about mathematics or algebra; I'm searching for some kind of application ...
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How does advancing through the math major work?

I am an undergrad math major that just completed Calculus 3 last semester. This semester I signed up for Discrete Mathematics, and will be taking Intro to Advanced/Abstract Math next. Of course-- I ...
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Using a Sylow Counting Argument

Let G be a group of order $$|G|=pq^m$$ where $p$ and $q$ are primes and $q^m<p$. Show that $$G\cong C_p \rtimes_h Q$$ where $Q$ is a group with $|Q|=q^m$ and $h:Q\rightarrow Aut(C_p)$ is a ...
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1answer
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Cartesian product to direct sum

I have no idea, how to prove rigorously the corollary from the proposition. I know that i can use the isomorphism $\phi:x_1e_1+...+x_me_m \in \oplus_i^mvect(e_i)\to (x_1e_1,...,x_m e_m) \in ...
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Show that no ring containing R can contain a root of g(x) = 3x +1

Show that if $R = \mathbb Z_6$ and $g(x) = 3x + 1 ∈ R[x]$, then $R[x]/(g(x)R[x])$ does not contain a root of $g(x)$. More generally, show that no ring containing $R$ can contain a root of $g(x)$. ...
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Field characteristic for a finite product of fields of characteristic $0$

Kind of a silly question, but is a finite product of fields of characteristic $0$ also of characteristic $0$? For instance, $\mathbb{C}$ has characteristic $0$, but then does $\mathbb{C}^n, n>1$ ...
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Ring of matrices has no nontrivial ideals [duplicate]

It is a theorem that a commutative ring is a field if and only if it has no nontrivial ideals. Clearly this does not hold in the noncommutative case. I am trying to show for instance that the ring of ...
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Reducibility over $Q$ implies reducibility over $Z$.

Let $f(x) \in$ $Z[x]$ , if $f(x)$ is reducible over $Q$ , then it is reducible over $Z$. I went through the proof from the book I'm reading , which starts as follows : We're given $f(x)$ is ...
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1answer
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Two non-isomorphic groups which are epimorphic images of each other

Let $G$ and $H$ be two groups, I am looking for an example such that $G$ is an epimorphic image of $H$ and $H$ is an epimorphic image of $G$ (i.e. they are both quotients of the other group), but they ...
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For which rings does a polynomial in $R$ have finitely many roots?

Which infinite rings satisfy the following Every non-zero polynomial in $R[X]$ has only finitely many roots ? Are there such rings which are not integral domains ?
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1answer
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Group of exponent $2$.

When I have a group $G$ of exponent $2$ and I know that all elements in $G-\{e\}$ are conjugated, is it right that $G$ is of order $2$? My try: For $g,h \in G - \{e\}$ the conjugation assumption ...
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1answer
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A $R$-module over a ring $R$ with identity is free iff it is a direct sum of copies of $R_R$, where $R_R$ is $R$ considered as a module over itself

Let $R$ be a ring with identity and $M$ be an $R$-right module. Then $M$ is called free over $X \subseteq M$ if for every module $N$ with mapping $\alpha : X \to N$ we can extend it uniquely to a ...
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Group of finite order where every element has infinite order

An often given example of a group of infinite order where every element has infinite order is the group $\dfrac{\mathbb{(Q, +)}}{(\mathbb{Z, +})}$. But I don't see why every element necessarily has ...
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Subgroup of square generators. [duplicate]

Let $N$ be a subgroup of $G$ with $N$ being generated by $\{x^2|x \in G\}$. Prove that $N$ is a normal subgroup of $G$. And that $[G, G] \subset N$. My idea was to look at $G/N$ and take $f:G ...
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What's an Isomorphism?

I'm familiar with the definition (inverses and bijections, preserving operations) in the context of groups and vector spaces, the hoeomorphism of topological spaces, and have some feeling for the ...
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Is/when is this property about the totative subset of $(\mathbb{Z}_n , +_n)$ true?

Is it possible to show (or when is it true); that for the group $\mathbb{Z}^+_n :=(\mathbb{Z}_n , +_n) $, there exists an $a \in \mathbb{Z}^+_n$ for each $z \in \mathbb{Z}^+_n$, where both $z+_n a$ ...
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Criterion for isomorphism of two groups given by generators and relations

When are two presentations of groups are isomorphic? In this post it is said: [...] find a set of generators of the first group that satisfies the relations of the second group [...] But I doubt ...