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.

learn more… | top users | synonyms (1)

0
votes
3answers
57 views

Inductive step in proof of Freshman's Dream

I am trying to prove that for $K$ a field of characteristic $p$ prime, $q$ a power of $p$ and $x,y$ in $K$, $$(x+y)^q=(x^q + y^q).$$ I have the base case, and now I am trying to do the inductive ...
0
votes
0answers
37 views

Alternate Proof of first (fundamental) isomorphism theorem

I've just been introduced to the fundamental isomorphism theorem and gone through the traditional proof that you show the ring isomorphism $\mu:R/\mathrm{ker}(\phi) \rightarrow S$ in which you define ...
2
votes
2answers
28 views

Let $G = \langle a \rangle$ be a cyclic group of order $n$. Show that for every divisor $d$ of $n$, there exists a subgroup $G$ whose order is $d$.

Let $G = \langle a \rangle$ be a cyclic group of order $n$. Show that for every divisor $d$ of $n$, there exists a subgroup $G$ whose order is $d$. If $d \mid n$ then there exists $m \in ...
1
vote
1answer
23 views

Cosets in quotient rings over ideals of the form $(n,f(x))$

When we take $\mathbb{Z}[X] / (n,f(X))$, $n \in \mathbb{Z}$ and $f(X) \in \mathbb{Z}[X]$, how do we construct the cosets? For example, consider the ideal $(2,X) = \{2p(X) + Xq(X) \mid p(X), q(X) ...
1
vote
0answers
39 views

Monoids in the real world? [on hold]

How are monoids helpful in the real world? I can see that they are a building block for the real field for example, but I don't see why we would need to show this break down of part of the real ...
3
votes
1answer
35 views

Reducibility of Representations over Finite fields

So, there are several standard ways of proving irreducibility/reducibility for representations over fields where the characteristic doesn't divide $|G|$ such as Maschke's theorem, jordan normal form, ...
0
votes
1answer
30 views

A quadratic form over $K-$vector space $V$

Let $K$ a field, $\operatorname{char} K \ne 2$. Definition: A quadratic form over $K$ is a homogeneous polynomial $Q(x_1, x_2, \dots , x_n) \in K[x_1, x_2, \dots , x_n]$ of degree $2$. If ...
0
votes
0answers
25 views

Specific Subgroups of an Abelian Group

I am looking for an elementary proof of the following result: If G is a finite abelian group and H is a subgroup of G, then G contains a subgroup isomorphic to G/H. This can be proved rather easily ...
0
votes
1answer
32 views

Let $f: G \to H $ be a group isomorphism. Show that for every $a \in G$, one has $o(f(a)) = o(a)$

Let $ o(a)$ denote the order of $ a $ in a group $ G $. Let $f: G \to H $ be a group isomorphism. Show that for every $a \in G$, one has $o(f(a)) =o(a)$ Proof. Assume that $o(a) = n$ and that $ o ...
0
votes
2answers
37 views

Determine the kernel of a linear map $f:U \to V$

Let $U=<\begin{pmatrix} 1 & 0 & 1 \\ 0 & 1 & 0\end{pmatrix}, \begin{pmatrix}0 & 0 & -1 \\ 0 & 1 & 0\end{pmatrix}, \begin{pmatrix}2 & 0 & 1 \\ 0 & 1 & ...
1
vote
2answers
39 views

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 ...
1
vote
3answers
38 views

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 ...
2
votes
2answers
46 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 ...
1
vote
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 ...
1
vote
4answers
46 views

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 ...
-1
votes
2answers
33 views

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 ...
2
votes
1answer
23 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 ...
0
votes
0answers
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 ...
2
votes
1answer
39 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 ...
1
vote
0answers
23 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 ...
0
votes
1answer
5 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 ...
0
votes
1answer
28 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 ...
0
votes
2answers
38 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. ...
2
votes
1answer
41 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 ...
0
votes
1answer
20 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 ...
1
vote
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 ...
4
votes
1answer
40 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 ...
1
vote
0answers
32 views

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 ...
1
vote
1answer
38 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 $E(\mathbb{Q})_{\text{torsion}}$ ...
1
vote
1answer
34 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 ...
0
votes
0answers
22 views

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 ...
1
vote
1answer
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 ...
0
votes
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 ...
2
votes
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 ...
0
votes
1answer
32 views

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 ...
0
votes
1answer
18 views

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 ...
1
vote
1answer
28 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 ...
1
vote
2answers
32 views

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 ...
2
votes
1answer
18 views

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 ...
1
vote
1answer
16 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 ...
1
vote
1answer
28 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 ...
0
votes
2answers
46 views
+100

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 $GL_2(\mathbb{F}_p)$ ...
1
vote
2answers
38 views

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 ...
4
votes
0answers
34 views

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 ...
2
votes
1answer
69 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] ...
0
votes
2answers
22 views

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} ...
1
vote
1answer
24 views

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, ...
0
votes
1answer
23 views

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 ...
2
votes
0answers
34 views

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 ...
0
votes
1answer
9 views

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 $x \Delta y \Delta x=x$, ∀$x,y \in \mathbb{E}$.