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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General form of an element of the othogonal basis of $q$

Let $$q \begin{pmatrix}a & b \\ c & d\end{pmatrix}= (a-b)^2+(b-c)^2+(c-d)^2$$ quadratic form on $M_2(\mathbb{R})$. How can I prove that every orthogonal basis $B$ of $M_2(\mathbb{R})$ has ...
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Diagonalizability of endomorphism $f:M_2(\mathbb{R}) \to M_2(\mathbb{R})$ such that $f(A)=tr(A)I_2-A$.

Let $f:M_2(\mathbb{R}) \to M_2(\mathbb{R})$ such that $f(A)=tr(A)I_2-A$. How can I determine what is the explicit expression of $f$, and, most importantly, how do I see if it is diagonalizable? The ...
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1answer
30 views

Do there exist pro-$p$ groups with finite quotients of non $p$ power order?

We define a pro-$p$ group to be a projective (i.e. inverse) limit of $p$-groups. My question is exactly as stated in the title: If a subgroup $H$ of a pro-$p$ group $G$ has finite quotient, must ...
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1answer
25 views

Clarifications of problem with parameters: the relationship between matrices and endomorphism

Let $f$ be an endomorphism of $R^3$ such that $f(a,b,c)=(2b,a-b,b)$. I don't understand how I can see for which values of $k\in R$ there esist $$\begin{pmatrix}-2 & 0 & 0 \\ 0 & k & 0 ...
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4answers
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Field Isomorphisms and Square Free Integers

I need to prove the following: Let $D$ be a square free integer. Show that $ \lbrace\begin{pmatrix} a & bD \\ b & a \end{pmatrix} \mid a,b\in\mathbb{Q}\rbrace $ is a field ...
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Torsion elements of a module not a subgroup

Consider $A$ a ring, $M$ an A-module, and $Tor(M)$ being the set of torsion elements of M (that is, the set of $m \in M$ for which $am=0$ for some $a \in A\backslash \{0\}$ ) Show that $Tor(M)$ need ...
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1answer
22 views

Inequivalent representations of a finite group

I'm looking for this result: A finite group has only finitely many inequivalent representations of given degree over a field of characteristic $0$. Do someone know where I can find a proof of ...
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34 views

There are $72$ non-trivial points

We have that $-2 \sqrt{2} < x < 2 \sqrt{2} \ \ $ : $5$ integers $-2 \sqrt{2} < y < 2 \sqrt{2} \ \ $ : $5$ integers $-2 < z < 2 \ \ $ : $3$ integers To find how many ...
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1answer
38 views

Computation rules for tensor products and inner products

I'm studying distributions and just came along the formula $$\langle f\otimes g,\phi\otimes\psi\rangle=\langle f,\phi\rangle\langle g,\psi\rangle.$$ I understand what that means in the context of ...
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22 views

Isomorphism between modules and submodules

Let $A$ be a ring, $M_1,\ldots,M_n$ be A-modules and $N_j$ be an $A$-submodule of $M_j$ for $1 \leq j \leq n$ Prove that $$(M_1 \times M_2 \times \cdots \times M_n)/(N_1 \times N_2 \times \cdots ...
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3 views

Associative and anticommutative Binary Operation(composition)

Show that if binary operation ,$\Delta$, is associative and anticommutative on $\mathbb{E}$, then $x\Delta y \Delta z=x\Delta z$ ∀$x,y,z \in \mathbb{E}$. [Hint: consider $x\Delta y\Delta z\Delta ...
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1answer
27 views

Let R be a ring and S be a subring of R with unity.

Let $R$ be a ring and $S$ be a subring of $R$. Suppose that $R$ does not have unity, but $S$ does. Let $1_S$ be the unity of S. Show that $1_S$ is a zero divisor of $R$. I've been stuck on this for ...
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34 views

$I$ is maximal ideal $\implies$ $R/I$ has no proper ideals

I'm reading through a proof in a book on commutative algebra and in the proof it uses the fact that $I$ is a maximal ideal $\implies$ $R/I$ has no proper ideals, by using the correspondence theorem. ...
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1answer
40 views

$g(x) | f(x)$ show that $(f(x)) \subset (g(x))$

I have been given a problem recently that has been puzzling me for some time. The problem states If $g(x), f(x)$ are elements of a polynomial ring $F[x]$ and $g(x) | f(x)$ show that $(f(x)) \subset ...
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1answer
19 views

Question on a fairly rigorous looking proof concerning the roots of a polynomial (resultants, symmetric polynomials, Viete)

Sorry for the big reading here. I tried to get as much on here so that it would make sense later on. Even though I put quite a bit on here, I actually just have one question about what is said ...
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1answer
37 views

Question about the proof of Hensel's Lemma

Hensel's Lemma: Let $F(x)=\alpha_0 + \alpha_1 x+ \dots + \alpha_n x^n \in \mathbb{Z}_p[x]$. We suppose that there is a $p$-adic $a_1 \in \mathbb{Z}_p$ such that $$F(a_1) \equiv 0 \mod p ...
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1answer
37 views

There is a primitive $m^{th}$ root of unity in $\mathbb{Q}_p$ $\Leftrightarrow m \mid (p-1)$

After Hensel's Lemma there is the following proposition in my notes: If $p$ is a prime and $m \in \mathbb{N}$ then there is a primitive $m^{th}$ root of unity in $\mathbb{Q}_p$ $\Leftrightarrow m ...
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Intuition behind the link between coding theory and group theory

I am trying to find an easy link between group theory and coding theory. The usual path that most of the texts follow is that they present introductory material on groups, fields, rings, etc., and ...
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1answer
34 views

Mazur's theorem-abelian torsion group of rational points of an elliptic curve

I am looking at Mazur's theorem... $$E(\mathbb{Q})_{\text{torsion}} \cong \mathbb{Z}/n\mathbb{Z}, \text{ for } n=1,2, \dots ,10,12$$ means that the torsion group $E(\mathbb{Q})_{\text{torsion}}$ ...
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1answer
29 views

The order of its elements in the additive group $\mathbb{Z}/9\mathbb{Z}$.

I want to determine the order of each element in the additive group $\mathbb{Z}/9\mathbb{Z}=\lbrace \bar{0},\bar{1},\bar{2},\dots, \bar{8}\rbrace$. It is taken from the Example (4) page 20 in Abstract ...
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Idea of Hensel's Lemma

$$P(x, y)=0 \tag {1} \\ \text{ where }P(x, y) \in \mathbb{Q}[x, y]$$ How do we know that $(1)$ has a solution in a $\mathbb{Q}_p$ ? We will apply Hensel's Lemma. Idea: we begin from an element ...
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26 views

$K-$rational solution of the equation - Is $\mathbb{Q} \leq \mathbb{Q}_p$?

Let $P(x, y) \in \mathbb{Q}[x, y]$. We consider the equation $P(x, y)=0$. If $a, b \in \mathbb{Q}$ such that $P(a, b)=0$ then $(a, b) \in \mathbb{Q}^2$, is called a rational solution. If $K$ a ...
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1answer
11 views

Issues with a particular bilinear form and determining rank, signature, etc. of its restriction

Let $b: M_2(\mathbb{R}) \times M_2(\mathbb{R}) \to \mathbb{R}$ such that $b(X,Y)=trace(X^tAY)$, where $X^t$ is the transpose of $X$ and $A=\begin{pmatrix} 2 & 1\\1 & 0\end{pmatrix}$. In my ...
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1answer
25 views

Kernel and diagonalizability of endomorphism $f:\mathbb{R_2[x] \to R_2[x]}$ such that $f(p)=p(1)x^2-p(k),$ for $k \in \mathbb{R}$.

Problem: Let $f:\mathbb{R_2[x] \to R_2[x]}$ be the endomorphism on the space of polynomials of degree less or equal than two such that $$f(p)=p(1)x^2-p(k),$$ for $k \in \mathbb{R}$. I have to ...
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what is the maximom order of an element is $\mathbb S_{15}$ [duplicate]

Let $S_n$ be the permutation group of order $n$. What is the maximom order of an element on $\mathbb S_{15}$. Is there any way to find maximom order of an element in $\mathbb S_{15}$ or find the ...
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Why $(f\mapsto f(v_i)w)_{i,j}$ with $f\in V'$,$w\in W$ is a basis of $\mathscr{L}(V',W)$?

I'm trying understand the proof of the Proposition 3.1.2 (pg.5) of this document: http://www.win.tue.nl/~amc/ow/lba/lba3.pdf Suppose $V$ and $W$ are finite dimensional. If $(v_i)_i$ is a basis of ...
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1answer
27 views

Restriction functors between Categories O over semi-simple Lie algebras

I have the following question: Let $\mathfrak{g}:= \mathfrak{gl}(3)$ be the general linear algebra and $\mathfrak{gl}(2) \cong \mathfrak{a} \subset \mathfrak{gl}(3)$ a sub-algebra. Let ...
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What does an ideal generated by a subset look like?

I am acquainted with principal ideals relatively well but I am struggling to understand the more general definition of ideals generated by an arbitrary subset of a ring. I suppose my question comes ...
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1answer
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Clarification of some doubts: working with the restriction of a quadratic form

Let $q:\mathbb{R^3}\to\mathbb{R}$ such that $$q(x,y,z)=2x^2+3y^2+4xy-2xz.$$ I have to determine rank and signature of $q$, and so far it should be fine: I got $\operatorname{rk}(q)=3$ and ...
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1answer
15 views

Infinite dimensional FG-modules

So the way I understand FG-modules is that it is analogous to a vector space defined over a field F with G a basis. However, I encountered a problem given the hypothesis that V is a possibly infinite ...
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1answer
27 views

Primitive element and field extension

If $K$ is an extension of field $F$ such that $[K:F]$ is finite and for two subfields $K_1$ and $K_2$ which contains $F$, either $K_2\subset K_1$ or $K_1\subset K_2$, then $K$ has a primitive element ...
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1answer
31 views

Classifying groups of order $2p^2$

I want to solve the following exercise from Dummit & Foote's Abstract Algebra text (p. 185 Exercise 15): Let $p$ be an odd prime. Prove that every element of order $2$ in $GL_2(\mathbb{F}_p)$ ...
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2answers
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Prove that $\langle x^3 + x + 1 \rangle$ is maximal in the polynomial ring $\mathbb{Z}_2[x]$

I'm assuming that there is an ideal properly containing this generated ideal and trying to show that this ideal contains $1$ and thus is equal to the $\mathbb{Z}_2[x]$. I've been multiplying various ...
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Can we find the GCD of two polynomials in $\mathbb Q[x]$ by representing the coefficients as vectors?

Can we find the GCD of two polynomials in $\mathbb Q[x]$ by representing the coefficients as vectors? For example: $f=x^5+3x^4+x^3+4x^2+1$, and $g=x^5+3x^4+4x^3+3x+1$ Can we represent these ...
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1answer
64 views

Finding all the group elements of a certain order of a finite group

Consider a group $G=\langle P, Z, Q \rangle$ generated $P,Z,Q$ where $$Z=\left[ {\begin{array}{ccc} -1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & -1\\ \end{array} } \right] ...
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Requirement That a Vector be Related to Itself Through Identity

If I have two vectors for which the relation can be written $$ \begin{bmatrix}\vec{I}_1\\\vec{I}_2\\\vec{I}_3\end{bmatrix} = \begin{bmatrix}A\end{bmatrix} ...
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1answer
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Let $f: \mathbb{R^3} \to \mathbb{R^3}$ with characteristic polynomial $-\lambda(\lambda-3)^2$ and $f(1,0,0)=(1,0,-1)$ and $f(0,1,0)=(2,3,1)$

Let $f: \mathbb{R^3} \to \mathbb{R^3}$ be a diagonalizable endomorphism with characteristic polynomial $-\lambda(\lambda-3)^2$ such that $f(1,0,0)=(1,0,-1)$ and $f(0,1,0)=(2,3,1)$. Given these data, ...
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Subring of a field extension is a subfield

For the first part, I am trying to show that any subring of $E/F$ is a subfield where $E$ is an extension of field $F$ and all elements of $E$ are algebraic over $F$. My solution is to just show an ...
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give a group that is isomorphic to the figure.

I think if I get help with one of these I should be good on the rest. 23) is concerned with figure a) It has one symmetry and 4 possible points. seems like it would have two elements that map to ...
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Associative Binary Operation(composition) is anticommutative iff idempotent…

if Binary Operation, $\Delta$, defined on $\mathbb{E}$ is associative, then $\Delta$ is anticommutative iff $\Delta$ is idempotent and $x \Delta y \Delta x=x$, ∀$x,y \in \mathbb{E}$.
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Lifting map between finitely generated modules

Suppose $A$ is a commutative ring with unit, and $M$ is a finitely-generated module with the surjection $\pi: A^n \twoheadrightarrow M$. Let $f : M \to M$ be a module homomorphism. I am trying to see ...
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56 views

What's the difference between these two definitions of polynomial function?

Definition 1: Given $a_n,...,a_1,a_0 \in \mathbb{R}$, a polynomial function is a function $p:\mathbb{R} \rightarrow\mathbb{R} $ such that $p(x)=a_nx^n+...+a_1x+a_0$ Definition 2: The function ...
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63 views

Order of the Rubik's cube group

Associated to the Rubik's cube is a group as described in this Wikipedia article: $G = \langle F, B, U, L, D, R\rangle$. For example, the element $F$ corresponds to rotating the front face clockwise ...
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25 views

I have a question about Viete's formulas

If I have a polynomial $a_n x^n + a_{n-1}x^{n-1}+ \cdots + a_1 x + a_0$, and the roots of the polynomial is $r_1,r_2,\ldots,r_n$, then I can rewrite the polynomial as, $a_n x^n + a_{n-1}x^{n-1} ...
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1answer
25 views

Does $G\to X$ have dense image iff $T_eG \to T_{\theta(e)}X$ is surjective?

Let $G$ be a connected algebraic group, $X$ a variety and and $\theta : G\to X$ be a morphism of varieties. (In particular it could be the orbit of some action of $G$.) Consider the corresponding ...
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3answers
34 views

$x^2+3$ has two zeros over ${\Bbb F}_p$ provided that $x^2+x+1\in{\Bbb F}_p[x]$ has two?

The following is an exercise in abstract algebra: If $p=1\pmod{3}$, then $x^2+x+1\in\Bbb{F}_p[x]$ has two zeros. Prove in this case that $-3$ is a quadratic residue mod $p$. Showing that ...
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1answer
38 views

Question about the ring of polynomials bounded on a real variety

Suppose that $I$ is a prime real radical ideal in the polynomial ring $\mathbb{R}[x_1,\ldots,x_n]$. "Real radical" means that if a sum of squares $a^2+b^2+\ldots$ is an element of $I$ then so are the ...
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1answer
20 views

Vector Spaces and Simple Modules

Let $G$ be a finite group and let $R = \textbf{R}[G]$ be the group ring of $G$ with coefficients in the field $\textbf{R}$ of real numbers. Let $V$ be an $R$-module which is finite-dimensional as an ...
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62 views

Infinite rings with lots of zero divisors

Today I was trying to find an infinite ring $R$ whose all nonzero and nonidentity elements were zero divisors and actually found one: $\mathcal R =\text{Fun}(\mathbb N, \mathbb Z/2\mathbb Z)$. Given a ...
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Algebra finite over a subalgebra

I'm afraid my question is quite stupid, but I can't find the definition I need: given $A$ a graded algebra, i would like to know what does it mean that $A$ is "finite" over a sub-algebra. In ...