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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$f\cdot g=0 \implies f=0 $ or $g=0$.

I know this is kind of an obvious thing to say: Let $f,g \in \Bbb K[x]$, then $$f\cdot g=0 \implies f=0 \text{ or } g=0$$ But to my surprise I couldn't prove it. What's a simple way to do this?
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1answer
22 views

Tensor Products of bimodules over commutative rings

Suppose that $ R $ and $ S $ are commutative rings with identity, $ R \subset S $, $ 1 _{R} = 1_{S} $, $ M $ is a $ (S,R)$-bimodule, $ N $ is a $ (R, S)$-bimodule, $ T = M ...
0
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1answer
35 views

Show that a group with 21 elements contains at max 3 subgroups with 7 elements

I don't know if lagrange's theorem apply here anyway. I just know that this group can have a subgroup with 3 elements, but I don't know about any theorem that talks about number of elements of the ...
3
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5answers
69 views

What is a set of bijections?

I am taking a course on abstract algebra, and the lector defined $T$ to be a set, and defined $G$ to be the set of all bijections from $T$ to itself: $$ G=\{\text{all bijections }g\colon T\rightarrow ...
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1answer
21 views

Motivic measure

Somebody can give me some good references for start to read Motivic-measure, Now I`m studing the Grothendieck Ring, and is necesary undertand something of motivic theory for my case, so I need a good ...
1
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0answers
32 views

What is the vector product $(x\wedge y)\wedge z$?

Here's an exercise from my book (exercise 10, chapter 2.1) Show that the three-dimensional vector space $V=R^3$ forms an associative algebra with respect to the operation $x\uparrow ...
1
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2answers
56 views

Computing Factor Group

I am reading John Fraleigh's First Course in Abstract Algebra, $\S$36 on the Second Isomorphism Theorem which says that if $H < G$ and $N \triangleleft G$, then $$(HN)/N \cong H/(H \cap N).$$ He ...
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1answer
16 views

Is Belnap's four valued-logic a boolean algebra?

Belnap's logic contains the the truth values 'true' (t), 'false' (f), 'unknown' ($\bot$) and 'paradox' (T). Each of these is represented by pair a of bits: t $\rightarrow$ (1,0) f $\rightarrow$ (0,1) ...
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0answers
13 views

boolean algebra - belnap logic

How to find out wether an algebra is a correct boolean algebra? So if we have the following algebra (rejects to belnap-logic theorems): $ \langle \{ w,f, \top , \bot \} , \wedge \vee \neg \rangle$
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1answer
22 views

Isomorphic image of simple algebra is simple algebra.

I would like to proof following theorem: if E is simple algebra (over field K) and F is an algebra (over K) and $h: E \to F$ is a isomorphism, then F is simple algebra. Of course zero ideal and ...
5
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1answer
42 views

Does there exist a prime number $p$ such that $p\mathcal{O}_{K}$ in $K=\mathbb{Q}(\sqrt{2},\sqrt{3})$ is a prime ideal?

Problem: Prove or disprove: there exists a prime number $p$ such that $p\mathcal{O}_{K}$ in $K=\mathbb{Q}(\sqrt{2},\sqrt{3})$ is a prime ideal, where $\mathcal{O}_K$ denotes the ring of algebraic ...
3
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1answer
30 views

Find the order of $\tau^{100}$

Let $\tau= \left( \begin{array}{ccc} ...
1
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1answer
19 views

Group action of $GL(2, F)$ on the projective line $P(F)$

I refer to section 8.3, page 119 of Algebra, A Computational Introduction by John Scherk. It is about group action of $GL(2, F)$ on the projective line $P(F) = F \cup \{\infty\}$. Given a matrix ...
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3answers
33 views

Cyclic Groups, find the generator [on hold]

Let $a$ be an element in a group $G$. What is a generator for the subgroup $H = G_1 \cap G_2$ where $G_1, G_2$ are the groups generated by $a^m, a^n$, respectively?
5
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0answers
28 views

Why is $Ind^G_H(M)=Ind^{G/H}_{\{e\}}$?

I was looking at some representation theory notes and found the following statement: $Ind^G_H(V)=\mathbb{C}[G]\otimes_{\mathbb{C}[H]}V=\mathbb{C}[G/H]\otimes_\mathbb{C} V$. Now, this makes intuitive ...
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2answers
37 views

Domain and primes ideals [on hold]

Let $A$ be a commutative ring with identity and $A[x]$ a polynomial ring. Show that the ideal $(x)$ is prime in $A[x]$ iff $A$ is domain.
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2answers
43 views

Why normal subgroup chains in Galois theory

I have began understanding Galois theory and I had a question regarding the relationships of normal subgroups to field extensions. So given an irreducible polynomial over the rationals $$a_1 + a_2x ...
2
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2answers
59 views

Is a polynomial $f$ zero at $(a_1,\ldots,a_n)$ iff $f$ lies in the ideal $(X-a_1,\ldots,X-a_n)$?

This is probably a very silly question: If $R$ is an arbitrary commutative ring with unit and $f\in R[X]$ a polynomial, then for any element $a\in R$ we have $$f(a)=0 \Longleftrightarrow X-a ~\mbox{ ...
3
votes
3answers
73 views

Show that $G/H\cong\mathbb{R}^*$

Let $G:= \bigg\{\left( \begin{array}{ccc} a & b \\ 0 & a \\ \end{array} \right)\mid a,b \in \mathbb{R},a\ne 0\bigg\}$ Let $H:= \bigg\{\left( \begin{array}{ccc} 1 & b \\ 0 & 1 \\ ...
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1answer
26 views

Proving property of group-like algebraic structures by means of induction

How do you prove (by means of induction) that the following is true for all group-like algebraic structures? $$\operatorname{ord}(a_1 \circ a_2 \circ a_3 \circ \cdots \circ a_{n-1} \circ a_n) = ...
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2answers
33 views

If $G$ is abelian, it has a subgroup in every order of $|G|'s$ divisors? [duplicate]

Assume that $G$ is an abelian group, I read somewhere that it can be derived from Lagrange's theorem that it has a number of subgroups that is equal to the number of $G$'s divisors. Why does it hold? ...
3
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1answer
42 views

Are there some theorems about the structure of a finitely generated $\mathbb Z[x]$-module?

There is a very clear picture about the structure of any finitely generated abelian groups. Are there some theorems about the structure of a finitely generated $\mathbb Z[x]$-module?
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2answers
36 views

Finding subgroups of $\mathbb{Z}_{13}^*$

I need to find all nontrivial subgroups of $G:=\mathbb{Z}_{13}^*$ (with multiplication without zero) My attempt: $G$ is cyclic so the order of subgroup of $G$ must be $2,3,4,6$ Now to look for ...
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1answer
40 views

performing a power operation ($a^n$) in a ring

In a ring - when performing a power operation, i.e $a^n$, to which operation is it related to? $+$ or $*$? On one hand - I know that a power is defined on multiplication - in "regular" numbers, but ...
2
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1answer
28 views

Count the number of monic irreducible polynomials of degree 12 over $\mathbb F_q$

This is a qualifying problem. I cannot understand how the inclusion exclusion principle work here in detail. However, I have an argument which leads to a different answer. I am not sure ...
4
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1answer
37 views

Number of Solutions to Polynomials in Finite Fields

Let $\mathbb{F}$ be a finite field and $f_i\in\mathbb{F}[x_1,x_2,\ldots,x_n]$ be polynomials of degree $d_i$, where $1\leq i\leq r$, such that $f_i(0,\ldots,0) = 0$ for all $i$. Show that if ...
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0answers
40 views

Definition of tensor product using pushout.

Let $M, N$ be right and left $A$-modules respectively. Then we have actions $$ \varphi_M: M \times A \to M, \\ \varphi_N: A \times N \to N. $$ There is a definition of $M \otimes_A N$ as follows. We ...
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1answer
36 views

$G$ finite, $P < G$ a Sylow p-subgroup, $N_{G}(P)$ the normalizer is contained in $H < G$, show $N_{G}(H)=H$

Let $G$ be a finite group and $P<G$ be a Sylow p-subgroup. Let $N_{G}(P)$ be the normalizer of $P$ in $G$. Let $H<G$ be a subgroup containing $N_{G}(P)$. Prove that $N_{G}(H)=H$. I've been ...
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3answers
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Is there an algorithm to compute the degree of a polynomial?

Let $f\in k[X]$ be a polynomial in one unknown over any field (or any nice enough commutative ring, I imagine - it shouldn't matter) and suppose that all we can do to understand $f$ is to evaluate it ...
6
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3answers
77 views

G/N read as G modulo N.

In my abstract algebra course, the instructor is calling G/N (the set of left Cosets of N in G) G mod N. This has not yet been explained. Why is this the case? My immediate suspicion is some ...
0
votes
1answer
19 views

A primitive element in a field of order $r$ is a primitive $(r-1)$st root of unity.

Could someone explain to me the following sentence? "A primitive element in a field of order $r$ is a primitive $(r-1)$st root of unity." Does this mean that for each element $x$ of a field of ...
1
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1answer
46 views

Prove that an $n$-dimensional non-unital algebra over a field $k$ is isomorphic to a subalgebra $\mathfrak{M}_n(k)$

Assume we have already proved this for unital algebras. Here's my book's solution: Construct the unital algebra $A^1$ [with unit $(1,0)$] as an algebra on the vector space $k\oplus A$ with the ...
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1answer
51 views

General Linear Group over the quaternions is a a topological group

How to show that General Linear Group over the quaternions is a a topological group?
2
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1answer
36 views

When is the tensor product commutative?

I am working with the the tensor product $-\otimes_R -$ over some noncommutative ring $R$. Is the tensor product always commutative if $R$ is commutative? If so, can the tensor product be commutative ...
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0answers
12 views

Involution on the set of all multipliers of $A$ ($A$ is a $C^*$-algebra)

Let $A$ be a $C^*$-algebra. $M(A)$ denotes the set of all multipliers of $A$, i.e. $m\in M(A)$ means that there is a map $m^*:A\to A$ such that $m(a)^*b=a^*m^*(b)$ for all $a,b\in A$. I want to know ...
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1answer
35 views

Splitting fields over $\mathbb{Q}$

Find a splitting fields over $\mathbb{Q}$ for: i)$x^4+4=(x^2-2x+2)(x^2+2x+2)$ (both factors are irreducible). The roots: $x_1=1+i,\ x_2=-(1+i)$. So the splitting field is $\mathbb{Q}(i)$, which has ...
3
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2answers
64 views

Subgroups of $\mathbb{Z}_{13}^*$ using “GAP language”

How can I know the nontrivial subgroups of $\mathbb{Z}_{13}^*$ (with multiplication without zero) using GAP language
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1answer
32 views

The quotient of a ring by the annihilator of an ideal

Let $R$ be a commutative ring with identity and $I$ an ideal of $R$. It's true that we have an $R$-module isomorphism $$I\cong R/ann_RI,$$ where $ann_RI=\{x\in R:xr=0,\;for\;all\;r\in I\}$ is the ...
3
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0answers
23 views

Question concerning a minimal faithful left ideal in an artin algebra

Let $B$ be an artin algebra an suppose there is a faithful projective-injective left $B$-module. Moreover, there is a minimal faithful left ideal $Be$ for some idempotent $e\in B$. 1) What does ...
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2answers
40 views

Show that $G:=\mathbb{Z}_{13}^*$ is cyclic

I need to prove that $G:=\mathbb{Z}_{13}^*$ (without zero with multipcation)is cyclic My attempt: I tried to check each element in $G$ if it is a generator or not: $$ \begin{align} &1^1=1\mod ...
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1answer
35 views

Fundamental Thm of Finite Abelian Groups proof

Let $G$ be a finite Abelian group of order $p^nm$, where $p$ is a prime that does not divide $m$. Then $G=H\times K$, where $H=\{x\in G|x^{p^n}=e\}$ and $K=\{x\in G|x^m=e\}$. Moreover,$|H|=p^n$. I ...
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1answer
34 views

Every irreducible polynomial f over perfect field F is separable

Every irreducible polynomial f over perfect field F is separable. Can you check my proof? Let f is inseparable. So we have $f=\sum_i h_ix^i$ and $f^p=\sum_i h_i^px^{ip}$ Now I use Frobenius mapping ...
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2answers
21 views

Characteristic is positive and exist polynomial with $g(x^p)=f$

$F$ is a field. $f \in F[X]$ is inseparable and irreducible. Show that characteristic p of F is positive and there exists $g$ with $g(x^p)=f$. We know that f is inseparable so $gcd(f,f')\neq 1$, so ...
2
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0answers
89 views

Why is $\sum a_i \exp(b_i)$ always equal to $0$?

Let $z$ be complex. Let $a_i,b_i$ be polynomials of $z$ with real coefficients. Also the $a_i$ are non-zero and the non-constant parts of the polynomials $b_i$ are distinct. (*) Let $j > 1$. ...
0
votes
2answers
70 views

Does such a Galois extension exist?

Let $K = \mathbb{Q}(\sqrt{-3})$, an imaginary quadratic field. Does there exist a finite Galois extension $L/\mathbb{Q}$ which contains $K$ such that $Gal(L/\mathbb{Q})$ is isomorphic to $S_3$? Here ...
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1answer
33 views

Determining the center of the p-Sylow subgroup of $S_p $

My Algebra book says without proof that the center of the p-Sylow subgroup (we will call it P) of $S_p$ is the subgroup itself. Now I can understand that P will be a subset of its center as it is of ...
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2answers
19 views

Find isomorphism for an operation

I was trying to solve this problem, but am having trouble seeing why it is an isomorphism. To map from R* to G, I think that the phi function would be Phi(x)=x/2 but that doesn't work. This phi ...
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3answers
35 views

What does it mean to evaluate a polynomial at a linear map, e.g. $p(T)$ where $p(x)$ is a polynomial?

I'm learning about the Cayley-Hamilton Theorem and the Primary Decomposition Theorem, and have trouble understanding what the notations mean. What does $\chi_T(T)$ mean? ($\chi_T(x)$ is the ...
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0answers
33 views

Direct sum notation

I was reading the direct sum of the groups and the index notation looks little bit strange for me. Group $G = \bigoplus_{\alpha < \beta}\mathbb Zx_{\alpha},$ where $\beta$ be an ordinal. Is this ...
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2answers
34 views

State a reason the given function is not a homomorphism

$f:\Bbb R \rightarrow \Bbb R$ and $f(x)=\sqrt x$ For $\forall x\lt0\in\Bbb R$, $f(x)=\sqrt x\in\Bbb C\notin\Bbb R$ Does my answer make sense, or should I elaborate with words?