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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$K/F$ Galois Extension, $H \leq G$, where $G$ the Galois group, then there is $\alpha \in K$ s.t. $H=\{\sigma \in G : \sigma \alpha = \alpha\}$.

Let $K/F$ be a finite Galois extension of fields with Galois group $G$. Let $H$ be a subgroup of $G$. Then there is $\alpha \in K$ such that $H=\{\sigma \in G : \sigma \alpha = \alpha\}$. My proof ...
4
votes
3answers
125 views

Every normal subgroup is the kernel of some homomorphism

Clearly the kernel of a group homomorphism is normal, but I often hear my professor mention that any normal subgroup is the kernel of some homomorphism. This feels correct but isn't entirely obvious ...
2
votes
0answers
52 views

Composition factoring into nonnegative integer polynomials

Consider the integer polynomials with nonnegative coefficients, such as: $$ 1 + 2x +2x^2 $$ $$ 3 + 3x^4 + 11x^{10}$$ $$ 13 + 7x + x^2 $$ I asm interested in knowing "what is a composition ...
-1
votes
1answer
48 views

True or false simple algebra questions (centralizers, conjugacy classes, normal groups, abelian groups) [closed]

Can someone please verify my answers to the following questions? Note: This is NOT homework! Answer true or false to the following questions: Two elements of a group in the same conjugacy ...
1
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0answers
30 views

Dihedral group and symmetry group of icosahedron

I am studying abstract algebra at the moment, but I have several troubles picturing the dihedral group $D_n$ and the symmetric group of the icosahedron. For instance, I find it really hard to solve ...
1
vote
1answer
30 views

Relating definitions of a normal field extension.

I have come across the following two definitions of a normal field extension. $\textbf{Definition 1:}$ An algebraic field extension $L/K$ is said to be normal if $L$ is the splitting field of a ...
0
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1answer
33 views

If $F$ is a field, then any two algebraic closures are isomorphic by an isomorphism that is the identity on $F$.

To start, suppose $K_1$ and $K_2$ are two algebraic closures of $F$. (a) Let $P$ be the set of partial functions $f$ from $K_1$ to $K_2$ with the following properties: $F$ is contained in ...
1
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1answer
20 views

Notation involving field extension

I am currently reading notes on Galois Theory and have come upon the following proposition, Let $f(x)$ be the minimal polynomial of a generator $\alpha$ of a finite field extension $k(\alpha)$ of ...
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0answers
22 views

$ G $ is satisfies the maximal permutizer condition if $ G $ is supersoluble?

Permutizer of a subgroup $ H $ of $ G $ is defined to be the subgroup generated by all cyclic subgroups of $ G $ that permute with $ H $, i.e. $ \langle x\in G \vert \langle x \rangle H = H \langle ...
0
votes
1answer
24 views

Semidirect Product of Two Groups

So I am beginning to learn Semidirect product. Now I have to identify the semidirect product of the two groups $Z_{p}\times Z_{p}$ =$H$ and $K$=$Z_{p}$ where p is an odd prime. So I can write ...
19
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3answers
397 views

Stanford math qual: Abelian groups $G$ satisfying $0\to \Bbb{Z} \oplus \Bbb{Z}/3\Bbb{Z} \to G \to \Bbb{Z} \oplus \Bbb{Z}/3\Bbb{Z} \to 0$

I am studying for my qualifying exams and came across the following question: Find all abelian groups $G$ that fit into an exact sequence $0\to \Bbb{Z} \oplus \Bbb{Z}/3\Bbb{Z} \to G \to ...
2
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1answer
52 views

Finding all ideals in a finite ring

Let $\mathbb F_2$ be the field of two elements. Consider the factor ring $$R=\mathbb F_2[x, y]/\langle x^2, y^2\rangle.$$ I want to find all ideals of $R$. Note that ...
1
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1answer
29 views

Equivalence of different prime factorizations in $\mathbb{Z}[\zeta_3]$

I'm reading that in the ring $\mathbb{Z}[\zeta_3]$, where $\zeta_3$ is the cubic root of unity, two prime factorizations of $4 = 2 \times 2 = (1 + \sqrt{-3})(1 - \sqrt{-3})$ are equivalent, because up ...
3
votes
1answer
65 views

Finding the Fixed Fields in the Galois Correspondence for the Splitting Field of $x^4-3x^2+4$ over $\mathbb{Q}$

I have found the Galois group of the polynomial $x^4 - 3x^2 + 4$ (see below), but I am not sure how to find the fixed fields in the Galois correspondence. The roots of the polynomial are $$\pm ...
1
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1answer
25 views

Prove socle is ideal

In any ring $R$ define the socle as the sum of all minimal right ideals of $R$. Say we have two minimal ideals $A,B$. If $a\in A,b\in B$, then $a+b$ is in the socle. If $x\in R$, then ...
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0answers
24 views

Question on primitive element for extensions of finite fields [duplicate]

I was given this question in field theory class which I am stumped on as after some effort I still cannot tackle. I am given a general natural number $n>1$ and am asked to prove or disprove: ...
2
votes
2answers
61 views

Ideal of polynomials vanishing on $\{(x,y): x^2+y^2=1, x \neq 0 \}$

I'm reading the book "Introduction to algebraic geometry" by Hassett, and in Chapter 3, after introducing the concept of the ideal of polynomials vanishing on a set $S$, the author gives some ...
2
votes
1answer
33 views

Another abstract algebra/field theory question

Suppose that $F$ is a field, $S \subseteq F^n$ and $I$ is an ideal in $F[x_1, \cdots, x_n] = F[\bar{x}]$. Define $$I(S) = \{ f \in F[\bar{x}]: f(\bar{s}) = 0, \forall \bar{s} \in S\}$$ and ...
2
votes
1answer
54 views

Number of abelian groups of order 108 [duplicate]

What is the number of abelian groups of order 108 upto isomorphism ? To answer this I wrote explicitly the possible abelian groups of order 108 as follows : $$\Bbb Z_{108}$$ $$\Bbb ...
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2answers
54 views

Cardinality of a ring obtained by quotienting $\Bbb Z[x]$

Let $R$ be the ring $\Bbb Z[x]/((x^2+x+1)(x^3 +x+1))$ and I be the ideal generated by $2$ in $R$. What is the cardinality of the ring $R/I$? 27 32 64 infinite Now I was thinking $R$ could be ...
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1answer
45 views

Infinite tower of algebraic extensions

For all I know, the following fact should be true: Consider an infinite tower of extensions $L_0 \subset L_1 \subset L_2 \subset \cdots$ such that $L_{i + 1} / L_i$ is algebraic for all $i \in ...
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3answers
28 views

Centre of matrix ring over skew field

Let $R$ be a semisimple ring. Show that $R$ is simple iff the centre of $R$ is a field. Book's solution: If $R$ is simple, it has the form $\mathfrak{M}_n(K)$ for a skew field $K$, and its ...
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1answer
42 views

Find the number of units in $M_2(\mathbb Z_p)$ where $p$ is prime. [duplicate]

Find the number of units in $M_2(\mathbb Z_p)$ where $p$ is prime. I want a detailed solution, not just the number. $M_2$ means matrix of order $2\times 2$. I know the defn of units. But how to ...
3
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0answers
35 views

Finding relations of variables

Suppose that \begin{align*} x&=t+t^{-1}+t^2s+t^{-2}s^{-1}+ts^{-1}+t^{-1}s-6\\ y&=t+t^{-2}+ts+s^{-1}-4\\ z&=t^{-1}+t^2+t^{-1}s^{-1}+s-4 \end{align*} Find a polynomial $P(x, y, z)=0$ ...
9
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0answers
75 views

What is the intuition behind Dirichlet's Class Number Formula? [closed]

As the title of the question suggests, what is the intuition behind Dirichlet's Class Number Formula being true? The Dirichlet Class Number Formula is$$h(\mathcal{O}_D) = -{1\over{D}} \sum_{n=1}^D ...
2
votes
0answers
60 views

Where does the term “Ring” come from in Algebra? [duplicate]

Group and Field make some sense to me, but I can't see why the structures that are closed under two binary operations would indicate "ring".
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0answers
33 views

The space of alternating multilinear maps and existence of a bilinear map [duplicate]

Before, I ask similar this. But here I change question settings since it was incomplete. I hope receive good ideas. Let $E$ be a finite dimensional vector space over field $\mathbb R$ with $E^*$ as ...
2
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0answers
74 views

Show that $\mathbb{Z}_4\rightarrow \mathbb{Z}_8\oplus \mathbb{Z}_2\rightarrow \mathbb{Z}_4$ is exact [duplicate]

I want to know whether $0\rightarrow \mathbb{Z}_4\stackrel{f}\rightarrow \mathbb{Z}_8\oplus \mathbb{Z}_2\rightarrow \mathbb{Z}_4\rightarrow 0$ is exact wrt group homomorphism under addition. Since ...
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2answers
26 views

Group acting on $X$ and element of normal subgroup $H$ fixes an element of $X$ implies $H$ fixes all of $X$

A group $G$ acts on a set $X$ transitively and a normal subgroup $H$ fixes a point $x_{0} \in X$, i.e. $h \cdot x_{0}=x_{0}$ for all $h \in H$. Show that $h \cdot x = x$ for all $h \in H$ and $x \in ...
0
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0answers
29 views

Determinant of the change of basis for fractional ideal

Let $A$ be a fractional ideal of some number field extension $K:\Bbb Q$. Let $\omega_1, \dots ,\omega_n$ be a $\Bbb Z$ basis for $\mathcal O_K$ and let $\alpha_1, \dots ,\alpha_n$ be a $\Bbb Z$ basis ...
2
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4answers
83 views

Subgroup of $\mathbb{Z}$ generated by two positive integers

An exercise from Aluffi's Algebra book. Let $m,n$ be positive integers and consider the subgroup $\langle m,n\rangle$ of $\mathbb{Z}$ they generate. As a subgroup of $\mathbb{Z}$ it will be equal ...
0
votes
1answer
29 views

linear combination of polynomial equal to zero

I have a trouble with the following question. Let $p,q \in K[x]$. There are polynomials $a(x),b(X) \in K[x]$ with $\deg(a) < \deg(q)$ and $\deg(b) < \deg(p)$ such that $$a(x)p(x) + b(x)q(x) = ...
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2answers
57 views

If $G$ is a group with order $99$, it is cyclic by Sylow (isn't it?). I want to find a generator.

I have seen an argument in a specific case where $g,h\in G$ with $ord(g)=9$ and $ord(h)=11$ are used to create a generator through $f:=g^xh^y$ where $x, y \in \mathbb{Z}$ with $1=x9+y11$. Is this ...
2
votes
3answers
146 views

Find what $A/S$ is isomorphic to, where $A = \Bbb Z_2[X_1,\dots,X_n]$ and $S=\langle X_1^2-X_1,\dots,X_n^2-X_n\rangle$

Find what $A/S$ is isomorphic to, where $A = \Bbb Z_2[X_1,\dots,X_n]$ and $S=\langle X_1^2-X_1,\dots,X_n^2-X_n\rangle$. I'm sorry but I don't have anything to add here. I've been trying it with ...
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0answers
35 views

One particular permutation of 5 elements

As for options 1. and 2. I have taken a few examples randomly and found them to be correct but could not generalize and I am clueless about options 3. and 4. The answer ...
0
votes
1answer
33 views

Duality and tensor product of the Lie algebra

I would like to know how to compute the tensor product of the matrices below and how to deal with duality of vector spaces. The vector space I concern is the Lie algebra $\mathscr{sl_2}$ with basis ...
5
votes
1answer
46 views

Cofibration necessarily has closed image?

I know how to show that if $i: A \to X$ is a cofibration, then $i$ is injective, and in fact a homeomorphism onto its image. My question is, must the image necessarily be closed? I've tried ...
1
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1answer
52 views

How to tell if a given equation is not a class equation of a group?

Which of the following cannot be a class equation of a group of order $10$? $1+1+1+2+5=10$ $1+2+3+4 =10$ $1+2+2+5 =10$ $1+1+2+2+2+2=10$ As I can say options 2. ,1. and 4. are not ...
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2answers
28 views

Prove the following statement about polynomials [closed]

show that $x-1$ divides $ax^2+bx+c$ in $\mathbb Z[x]$, if $a+b+c=0$
0
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2answers
20 views

field trace and product

Let $a,b$ be elements in a number field. If the field (Galois) traces of $a,b $ are nonnegative, then is the field trace of $ab $ also nonnegative? Can I find an explicit formula of $Tr (ab) $ from ...
4
votes
2answers
48 views

Showing that the Field Extension $\mathbb{Q}(T^{1/4})/ \mathbb{Q}(T)$ is not Galois

Prove that $\mathbb{Q}(T^{1/4})$ is not Galois over $\mathbb{Q}(T)$, where $T$ is an indeterminate. I am not sure how to proceed due to the indeterminate. It suffices to show that the degree of ...
3
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1answer
51 views

Dummit & Foote's Abstract Algebra, 3rd Edition, Exercise 0.1.7

I feel awkward about my reasoning. Is it sound? Let $f:A\to B$ be a surjective map of sets. Prove that the relation $$a\sim b\quad\text{if and only if}\quad f(a)=f(b)$$ is an equivalence ...
0
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0answers
36 views

How do I prove that primary ideals satisfy this property?

Let $R$ be a commutative ring. Let $Q$ be a primary ideal of $R$. Let $I,J$ be ideals of $R$ such that $IJ\subset Q$. How do I prove that $I\subset Q$ or $J^n\subset Q$ for some positive integer ...
2
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1answer
30 views

Find polynomials $q(x)$ and $r(x)$ such that $f(x)=g(x)q(x)+r(x)$ where $r(x)=0$ or $\deg r(x)<\deg g(x)$

Find polynomials $q(x)$ and $r(x)$ such that $f(x)=g(x)q(x)+r(x)$ where $r(x)=0$ or $\deg r(x)<\deg g(x)$ provided that $f(x)=2x^4+x^2-x+1$ and $g(x)=3x^2+2$ in $\Bbb Z[x]$. The problem I'm ...
14
votes
2answers
175 views

Geometric reason as to why $H^2$ of the Klein bottle is $\mathbb{Z}/2\mathbb{Z}$?

I was reading this document when I came across the following: Recall that $H^2(K; \mathbb{Z}) \cong \mathbb{Z}/2\mathbb{Z}$. Here $K$ denotes the Klein bottle. Is there a good geometric ...
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0answers
16 views

What does it mean an ideal is nilpotent modulo another ideal?

Reference:Chatters, A. W.; Hajarnavis, C. R. (1971), "Non-commutative rings with primary decomposition", Quart. J. Math. Oxford Ser. (2) 22: 73–83, doi:10.1093/qmath/22.1.73 Let $R$ be an rng and ...
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votes
1answer
28 views

How do I prove that the standard definition of prime ideal is equivalent to that of Krull's? [duplicate]

Definition Let $R$ be a commutative ring and $I$ be a proper ideal of $R$. Then $I$ is prime if and only if $\forall a,b\in I, a\in I$ or $b\in I$. Let $R$ be a commutative ring and $P$ be a ...
3
votes
3answers
47 views

Find $\gcd(2x+7, x^2-2)=d(x)$ where all polynomials are in $\Bbb Q[x]$.

So, I used the Euclidean Algorithm to solve for the GCD: $x^2-2=(2x+7)(\frac12x-\frac74)+\frac{41}4$ $2x+7=\frac{41}4(\frac8{41}x+\frac{28}{41})+0$ $\therefore \gcd(x^2-2,2x+7)=\frac{41}4$ ...
2
votes
0answers
53 views

A field F can be extended to one in which all polynomials over F have a solution

Please check to see if my methods are correct. This question is a lot to absorb all at once. Let P be the set of non-constant polynomials over a field F and let X be the set of finite subsets of ...
0
votes
1answer
38 views

Abel-Ruffini implications

There are similar questions already regarding this topic but I'm hoping the answers can be diluted for me here. So regarding the fact that the roots of 5th order polynomials cannot be expressed as ...