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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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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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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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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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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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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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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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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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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39 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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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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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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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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$x^2+3$ 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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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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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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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 ...
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Isomorphism of quotient rings of $\mathbb{Z}[x]$

$\mathbb{Z}[x]/(x^2-a)$ is isomorphic to $\mathbb{Z}[x]/(x^2-b)$. Which $a$ and $b$ must be? Of course when $a$ equal to $b$, but I can't find any others. I tried to divide $(x^2-a)$ to ...
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Intermediate field extension

Consider de field $\mathbb F_2$ of two elements. Let $\theta$ be a zero of the irreducible polynomial $t^4+t+1$ and consider the extension $\mathbb F[\theta]$. May question is: Can we find an ...
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Show that $G$ contains elements $a,b$ s.t. $a^2=b^3=e$ and $ aba=b^2=b^{-1}$

Given that $|G|=6$ and is not commutative. I've showed that generators of $G$ have periods either $2$ or $3$, since $|H_a|=\text{period of generator a}$ divides $|G|$. Since $G$ is not commutative it ...
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If $a+bi$ is in $E_k$ then $a-bi$ is also in $E_k$?

I'm currently studying the properties of the Motzkin sets $E_k$, $k\in\mathbb{N}\cup\{0\}$ of the ring $\mathbb{Z}[i]$. The definition of $E_k$ is as follows: $E_0=\{0\}$, $E_1=$units of ...
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Do we have $\mathbb{C}[T] = \mathbb{C}[\Lambda] = \oplus_{\lambda \in \Lambda} \mathbb{C}[\lambda]$?

Let $\mathbb{C}[T]$ be the coordinate ring of a torus $T$. Suppose that $T$ acts on some variety $X$. Then $T$ acts on $\mathbb{C}[X]$: $t(f) = \lambda(t)f$ ($f$ is a homogenous function on ...
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Number of subgroups and normal subgroups

I am struggling to understand how to calculate the nunmber of subgroups with permutations, for example: How many normal subgroups does S3 have? How many subgroups of order 4 has group S4? And does ...
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Conditions equivalent to Noetherianness

Let $R$ be a left Noetherian ring. We know that any direct sum of injective left $R$-modules is again injective. Since any injective module is quasi-injective, we infer that (1):"any direct sum ...
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Algebra and homomorphism

Is there a homomorphism between each pair of algebras of the same type? Is there an infinite algebra that has only one subalgebra?
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Find orbit of $1$ for $\sigma$

$\sigma = \left( \begin{array}{cc}1&2&3&4&5&6\\3&1&4&5&6&2\end{array}\right)$ $ 1 \mathop{\rightarrow}^{\sigma} 3 \mathop{\rightarrow}^{\sigma} 4 ...
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If $ f:G \to H $ is a homomorphism between groups $ G$ and $ H $ and if $ X \subseteq G $ then $ f (\langle X \rangle) = \langle f (X) \rangle $.

Let $\langle X \rangle$ denote the subgroup generated by $ X. $ If $f: G \to H$ is a homomorphism between groups $G$ and $H$ and if $ X \subseteq G $ then $ f (\langle X \rangle) = \langle f (X) ...
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Permutation to a power $\sigma^{100}$

$\sigma = \left( \begin{array}{cc}1&2&3&4&5&6\\3&1&4&5&6&2\end{array}\right)$ I need to calculate $\sigma^{100}$ $\sigma = (1,2,3,4,5,6)$ has order 6, and ...
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Semisimple module example

I need to find an example of a module over $\mathbb{F}[x]$ which is two dimensional over the field $\mathbb{F}$ and not semisimple. I do not know how to do it. Thanks
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Show the union of two subrings is generally not a subring

Show that the union of two subrings is a subring if and only if either of the subring is contained in the other. I have no trouble in going from right to left but cannot seem to be able to go from ...
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1answer
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compute $\left|<\tau^2>\right|$ for the given permutation

$\tau = \left( \begin{array}{cc}1&2&3&4&5&6\\2&4&1&3&6&5\end{array}\right)$ I need to compute $|\langle \tau^2\rangle|$ I know $\tau^2 = \left( ...
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Polynomial with n real roots

Let $P(x) = x^n + a_{n-1}x^{n-1} + \cdots + a_{1}x + 1$ where $a_i$ are nonnegative and real. Assume $P$ has $n$ real roots. Prove $P(2) \geq 3^n$. I thought I had a good idea about ...
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When are two direct products of groups isomorphic?

I was thinking about the following problem: Suppose that $G_1 \cong G_2$ are isomorphic groups. Under what conditions on the groups $H_1,H_2$ will we have $$G_1 \times H_1 \cong G_2 \times H_2 ?$$ ...
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Simple and semi-simple over $\mathbb{Z}$

What is a necessary and sufficient condition on an integer $n$ for $\mathbb{Z}/n \mathbb{Z}$ to be simple as a module over $\mathbb{Z}$? Semisimple? In the case of simple I think that because it is a ...
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Showing that $x \ast y := x + y - \lfloor x + y \rfloor$ defines a group structure on $[0, 1)$

Define $x\ast y = x+y - \lfloor x+y \rfloor$ where $\lfloor \cdot \rfloor$ is the floor function. How do I prove that $([0,1),\ast)$ is a group? I was trying to separate cases and prove this, but it ...
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Find closed form for a sequence using the Fibonacci and Lucas number sequences. [on hold]

Define $A_n$ as follows: $$\begin{align*} A_0 &= 6 \\ A_1 &= 5 \\ A_n &= A_{n - 1} + A_{n - 2} \; \textrm{for} \; n \geq 2. \end{align*}$$ There is a unique ordered pair $(c,d)$ such that ...
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52 views

Galoisgroup of a polynomial

Task: Determine the galois group of $x^4-4x^2-11$ over $\mathbb Q$ The roots are obviously $\pm\sqrt{2\pm\sqrt{15}}$ But I have problems in checking whether just $\sqrt{2+\sqrt{15}}$ is generating ...
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Let $G$ be a group and $u \in G$ be a fixed element. By the following, prove that $(G,\bullet)$ is a group.

Let $G$ be a group and $u \in G$ be a fixed element. Define the operation $\bullet$ on G as $\forall a,b \in G, a \bullet b=au^{-1}b.$ Prove that $(G,\bullet)$ is a group. So, I know that in order ...
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Primitive polynomials: some statements to (dis-)prove

Prove or disprove: i) The sum of primitive polynomials in $\mathbb{Z}[x]$ is primitive ii) The product of primitive polynomials in $\mathbb{Z}[x]$ is primitive iii) There is only a ...
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Generated field is the same as composite field

I am considering Exercise 13.2.6 from Dummit & Foote's Abstract Algebra: Prove directly from the definitions that the field $F(\alpha_1, \alpha_2, \dots, \alpha_n)$ is the composite of the ...
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$\mathbb C,$ isomorphism to $\mathbb{R} \times \mathbb{R}$ under multiplication

How can I show, that $(\mathbb C,.)$ is not isomorph to $(\mathbb{R},.) \times (\mathbb{R},.)$ under multiplication? I tried to point out that $f(1) = 1$, then pair $(1,1) \rightarrow 1 + 0i$, but ...
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1answer
66 views

The diophantine equation $y^2=x^3+7$ has no solutions.

In my lecture notes there is the following example: The diophantine equation $y^2=x^3+7$ has no solutions. Proof: If the equation would have a solution, let $(x_0, y_0)$, $y_0^2=x_0^3+7$, then ...
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$a^2+ab+b^2=c^2+cd+d^2$ prove that $a+b+c+d$ is a composite number for positive integers $a,b,c,d$ [on hold]

(Positive integers $a,b,c,d$ meet this condition $a^2+ab+b^2=c^2+cd+d^2$ )prove that $a+b+c+d$ As in the topic my proof looks like that; $(a+b)^2 - ab=(c+d)^2-cd$ $(a+b)^2 - (c+d)^2=ab-cd$ ...