Abstract algebra is the study of algebraic objects. Some of the more common algebraic objects are groups, rings, fields, vector spaces, modules, and other advanced topics.

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How to show this ideal of polynomial ring contains the discriminant? [on hold]

Let $f=ax^2+bx+c$ with $a,b,c$ in a ring $R$. Show that the ideal of the polynomial ring $R[x]$ that is generated by $f$ and $f'$ contains the discriminant, the constant polynomial $b^2-4ac$.
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48 views

abstract algebra question with p-subgroup

Let $G$ be a finite group, $H<G$ a subgroup. Suppose that $H$ is a $p$-group, $p$ a prime, and $p\mid[G:H]$. Show that $p\mid[N_G(H):H]$ and $H<N_G(H)$. In particular, if $G$ is a $p$-group and ...
3
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1answer
48 views

Vector Spaces and Groups

I've just completed a course in linear algebra. I'm a physics undergraduate and I don't plan on taking an abstract algebra course. That said, I've been reading a little bit about it. As I understand ...
1
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1answer
48 views

Explanation for the definition of monomials as products of products

I'm attempting to learn abstract algebra, so I've been reading these notes by John Perry. Monomials are defined (p. 23) as $$ \mathbb{M} = \{x^a : a \in \mathbb{N}\} \hspace{10mm} \text{or} ...
2
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3answers
43 views

How can I find the degree of the extension?

Let $\omega_7=e^{2\pi i/7}$ . How can I find the degree of the extension $\mathbb{Q} \leq \mathbb{Q}(\omega_7+\omega_7^5)$?? Could you give me some hints??
2
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0answers
61 views

Localization of euclidean ring is euclidean?

I am trying to prove that a localization of a euclidean ring is euclidean, and the converse statement. I feel the basic definition of the norm is enough but I do not know how. Please note I am very ...
4
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2answers
45 views

Question about kernel and homomorphism

I was wondering is there any reason we take the identity e` for the kernel for ring homomorphism to be the additive identity instead of the multiplicative one?
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1answer
58 views

Factor $17i$ into a product of irreducible elements in $Z[i]$

Messing with some algebra I've got: $17i = (17+17i)(0.5+0.5i)$ I'm pretty sure this is not how I'm supposed to go about it. Can someone point me in the correct direction?
6
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3answers
62 views

Show that $\mathbb Q[x]/(x^2+2)$ and $\mathbb Q[x]/(x^2-2)$ are not isomorphic.

I have a proof that says $\mathbb Q[x]/(x^2+2)$ and $\mathbb Q[x]/(x^2-2)$ are not isomorphic. However I feel that it is not good one... First I see that $x^2+2$ and $x^2-2$ are irreducible in ...
1
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2answers
32 views

Understanding the definition of tensor product as a quotient of a free abelian group

I've been give the Definition: Let F be a free abelian group with a basis $X$ such that. $$F = \langle A\times B\mid \emptyset \rangle $$ Let $f$ be a subgroup of $F$ generated by the ...
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2answers
57 views

A silly question about $\mathbb{Z}_1$

Is the cyclic group of one element, i.e. $\mathbb{Z}_1$, just $\mathbb{Z}$? I would think not, since the integers have more than one elements. Or is it just $\langle 1\rangle$, the abelian group ...
2
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1answer
65 views

How many distinct integer solutions does the inequality $|x_{1}|+|x_{2}|+…+|x_{n}| \leq t$ have?

How many distinct integer solutions does the inequality $|x_{1}|+|x_{2}|+...+|x_{n}| \leq t$ have? We know that: $x_{i} \in Z,\ \forall i \ 0\leq i \leq n \ and \ t\geq0.\ $ I know that if we ...
1
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1answer
21 views

Irreducible in $F[x]$ implies irreducible in $R[x]$?

Let $R$ be a unique factorization domain, let $F$ be the field of fractions of $R$ and let $f(x)\in R[x]$. I want to show that if $f(x)$ is irreducible in $F[x]$ then $f(x)$ is irreducible in $R[x]$. ...
0
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1answer
49 views

Nakayama's lemma, second version

Let $R$ be a commutative ring with identity, $J$ an ideal that is contained in every maximal ideal of $R$, and $A$ is finitely generated $R-$ module. If $R/J\otimes _R A=0$, then $A=0$. ...
1
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1answer
17 views

If $F$ is algebraically closed and $tr.d.F/K$ is finite, then every $K$-monomorphism $F \rightarrow F$ is an automorphism

If $F$ is algebraically closed and $tr.d.F/K$ is finite, then every $K$-monomorphism $F \rightarrow F$ is an automorphism First, suppose that a transcendence base of $F$ over $K$ is $\{s\}$. Then ...
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0answers
26 views

Noetherian local ring with maximal ideal $M$

Let $R$ be a Noetherian local ring with maximal ideal $M$. If the ideal $M/M^2$ in $R/M^2$ is generated by $\{ a_1+M^2, \dots, a_n +M^2\}$, then the ideal $M$ is generated in $R$ by $\{ a_1, \dots , ...
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2answers
32 views

Clarification on quadratic ring notation

My Abstract Algebra text is using the notation $\mathbb{Z}[1 + \sqrt{-5}]$ and calling it a "quadratic integer ring." Just to clarify, $\mathbb{Z}[1 + \sqrt{-5}]$ is simply the set $$ \left\{ a + b(1 ...
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2answers
45 views

Confused on notions of maximal ideal and some notation

I'm just getting started learning ring theory and am currently learning about ideals. By book (Dummit & Foote) says the following: For example, in the ring $R = \mathbb{Z}[x]$ the elements $2$ ...
2
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1answer
20 views

Isomorphism between modules over a semisimple ring

If $P$ is a module over the semisimple ring $R/J$, where $R$ is a semilocal ring having $1$, and $J$ is its Jacobson radical, does any isomorphism $P⊕...⊕P≅P'⊕...⊕P'$ with the same (finite) number of ...
3
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0answers
49 views

what is the “largest” abelian subgroup of SL(2,Z)?

I suppose "largest" is a bit of a nebulous term, so to make it precisely, I suppose I could ask for the largest size of a minimal generating set of an abelian subgroup's image in $PSL(2,\mathbb{Z})$. ...
0
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0answers
14 views

From progenerators to progenerators

I know that if $R$ is a ring (with identity) and $P$ is a progenarator right $R$-module (a f.g. projective generator) then $P/PJ$ is clearly f.g. when $J$ is the Jacobson radical of $R$; but, how ...
2
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1answer
38 views

Are finitely presentable modules closed under extensions?

If $0 \to A \to B \to C \to 0$ is an exact sequence of modules, and $A$ and $C$ are finitely presentable, then is $B$ finitely presentable? The answer is "yes" if we replace modules with groups, ...
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3answers
53 views

Find the order of the intersection of two cyclic groups

Suppose $G=\langle a\rangle $ has order $140$. What is the order of the intersection $\langle a^{100}\rangle$ and $\langle a^{30}\rangle$? My attempt: I know that the order of $\langle ...
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1answer
19 views

About finite field extensions and their generators

I am with trouble to prove this statment: "Suppose that $\{\beta_1,...,\beta_n\}$ is a base of $L|K$ and $\mathcal{M}$ is a subset of some fied $M\supseteq L$. Prove that $\{\beta_1,...,\beta_n\}$ ...
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1answer
26 views

Centralizer, Normalizer and Conjugate

I am looking at Group Theory notes on Centralizer and Normalizer for next semester and come up with this question: Let $H$ be a subgroup of $G$, and let $g$ be an element in $G$. Show that $$(a)\ ...
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1answer
25 views

one to one correspondence of Ideals in a ring and its localization

Let $A$ be a commutative ring, and $S$ a mutiplicatively closed subset. In my text book, it is stated that: there is one to one correspondence of prime ideals in ring $A$ (not meeting $S$) and ...
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1answer
68 views

Subfields of $\mathbb Q(e^{\frac{i\pi}{4}})$

What are the subfields of $\mathbb Q(e^{\frac{i\pi}{4}})$ ? Since $\displaystyle e^{\frac{i\pi}{4}}=\frac{\sqrt2+\sqrt{-2}}2$ So $\mathbb Q(\sqrt2), \mathbb Q(\sqrt{-2})$ and ...
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37 views

If Zp is a field, when p is prime, does the multiplicative group F^x have a subgroup isomorphic to Zp x Zp?

I know Zp x Zp is not a field and I'm guessing there isn't a subgroup in F^x isomoprhic to Zp x Zp, due to the non-uniqueness of elements in Zp x Zp. But, I'm drawing a blank as to how to write it ...
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10 views

Is it ever possible for hypercomplexes to generate every element modulo a prime?

To start, we can take a well-chosen complex number, modulo a prime $p$, and generate every complex element modulo $p$. For example, if we take $(1+2i)^k \pmod 3$, each power of $k$ up to $(3 \cdot 3 ...
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3answers
136 views

Prove a quotient group is abelian

Let $G$ be a group with a normal subgroup $M$ such that $G/M$ is abelian. Let $N\geq M$ and $N \unlhd G$. Show $G/N$ is abelian. My attempt: To show that $G/N$ is abelian, we need to show that for ...
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0answers
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Finding all groups with this property. (Elementary Algebra)

Suppose G is a finitely generated group and for any 3 subgroups of G at least 2 of them are comparable. Find all Groups with this property. I was found this problem on web today and It seems nice to ...
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26 views

polynomial algebra and multiplications of its elements

According to the definition of the polynomial algebras $A(n)$ and $A(n,m)$ for n∈N and $m∈N^n$, if F be field GF(2) and $X_1,...,X_n$ be n pairwise commuting indeterminates over GF(2), then for a∈N we ...
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Show the intersection of a nonidentity normal subgroup and the center of P is not trivial

P is p-group and M is a nontrivial normal subgroup of P. Show the intersection of M and the center of P is nontrivial. By the class equation, I proved that Z(P)is not 1. Then, how do prove I the ...
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Question about geometrical invariant

Assume $R$ is ring and $I $is ideal of $R $ The property of ideal $I$ was defined Geomerical properties which only depend on radical of $I$ For example varieties and projective varieties with ...
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55 views

Finding conjugacy classes and normal subgroups of $D_8$, the dihedral group of order $16$ [duplicate]

What are the conjugacy classes for the dihedral group $D_8$ of order 16? What are its subgroups of order $4$, and which of them are normal subgroups? I know that $\{e\} ...
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1answer
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Intersection of distinct maximal ideals in a commutative ring with identity.

If $R$ is a commutative ring with identity and $M_1, \dots, M_r$ are distinct maximal ideals in $R$, then show that $M_1\cap M_2 \cap \cdots \cap M_r = M_1M_2\cdots M_r$. Is this true if "maximal" is ...
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4answers
36 views

Polynomials and the fundamental theorem of algebra

Let $p(z) = z^n + a_{n-1} z^{n-1} + ... + a_0 = 0 $ be a polynomial. first, I want to show that there exists numbers $\alpha, \beta $ such that $|z| > \beta \implies |p(z)| > \alpha $ I am ...
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1answer
30 views

Elements of $\mathbb{Q}(e)$

Is $e^n$ for $n$ an integer an element of the field $\mathbb{Q}(e)$?
3
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1answer
20 views

show a group with prime order product is solvable

Is a group with order $16*17$ solvable? I know that from Burnside this is solvable since 2 and 17 are prime and 4 is greater than 0. However, I am not allowed to use it, so what should I do? ...
2
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2answers
35 views

$R$ is a ring. Prove that $R/(0_R)\cong R$

$R$ is a ring. Prove that $R/(0_R)\cong R$. I don't quite understand what $R/(0_R)$ looks like. By definition of quotient ring, it should have cosets $a+(0_R)$ where $a\in R$. So $R/(0_R)$ and ...
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8answers
358 views

Every infinite abelian group has at least one element of infinite order?

Is the statement true? Every infinite abelian group has at least one element of infinite order. I am searching for an infinite abelian group with all elements having finite order. Please help me ...
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2answers
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Is Orzech's generalization of the surjective-endomorphism-is-injective theorem correct?

In math.stackexchange answer #239445, Makoto Kato quoted a statement from the paper Morris Orzech, Onto Endomorphisms are Isomorphisms, Amer. Math. Monthly 78 (1971), 357--362. The statement ...
2
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1answer
19 views

Cyclic congruencies

Suppose $a$ and $b$ are positive integers. Set $x_n := a^n $ modulo $b$. Consider $\{x_n\}$. My question is: is it always true that this sequence must be cyclic? I am guessing there is some ...
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1answer
20 views

Prove that if $M$ is an $R-$ projective left module then $M/IM$ is an $R/I-$ projective left module. [duplicate]

Let $I$ ba a two-sided ideal of a ring $R$ and $M$ be an $R-$ left module. Prove that if $M$ is an $R-$ projective left module then $M/IM$ is an $R/I-$ projective left module. It is easy to see that ...
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1answer
24 views

Subcategory of sets with surjective mapping [on hold]

Im new at algebra and Im trying to prove that a subcategory of Set category , where the objects are sets and morphisms are surjective mappings is really a subcategory and it is not full. The first ...
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1answer
14 views

Let $M$ be an $R-$module and $x\in M\setminus\left\{ 0\right\} $. Prove that there exists a left ideal of $R$, say $I$ such that $Rx\cong R/I $.

Let $M$ be an $R-$module and $x\in M\setminus\left\{ 0\right\} $. Prove that there exists a left ideal of $R$, say $I$ such that $Rx\cong R/I $. Help me some hints. Thank you in advance.
2
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1answer
57 views

Are there 2p elements of order p in $\mathbb{Z}_p \times\mathbb{Z}_p$?

I know the order of $\mathbb{Z}_p \times\mathbb{Z}_p$ is $p^2$, that there are $p^2$ elements in the set represented by $\mathbb{Z}_p \times\mathbb{Z}_p$. Of those elements how many are of order ...
2
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1answer
29 views

Cyclotomic extension of $\mathbb{F}_p((T))$

I feel very confused about why adding n-th roots of unity to $\mathbb{F}_p((T))$ would give $\mathbb{F}_{p^n}((T))$. (Is this true?)
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40 views

Two properties related to semisimple rings

Let $R$ be a semisimple ring Show the following (i) If $xy=1 \in R$, then $yx=1$. (ii) If $x \in R$ is such that $xR$ is a left ideal of $R$, then $xR=Rx$. I am pretty lost with the two items. I ...
2
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2answers
56 views

Find idempotents in $\Bbb Q[x]/\langle x^2 - 1\rangle$

I know that in $\Bbb Z[x]/\langle x^2 - 1\rangle$ the trivial idempotents are $0,1$ and the other idempotents are those elements in $\Bbb Z[x]$ that have the remainder $0$ or $1$ when divided by $x^2 ...