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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Polynomial algebras

What is a polynomial algebra? I cannot find any definition of this concept. Is it a set of n-ary polynomial symbols defined over an algebra, as defined on p.29 of this document? ...
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Free lattice in three generators

By general results for every set $X$ there is a free bounded lattice $L(X)$ on $X$. I would like to understand the element structure of this lattice. The cases $X=\emptyset$, $X=\{x\}$ and $X=\{x,y\}$ ...
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The relationship between subfields and subgroups of a finite field.

I am trying to get my head around the structure $GF(p^n)$ when viewed as a vector space of dimension $n$ over $GF(p)$ (mainly the relationship between the additive and multiplicative structures). I'm ...
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order and cycles of perfect shuffle of 52 cards

This is the shuffle: $$1,2,\cdots,52$$ is turned into $$1,27,2,28,\cdots,26,52$$ when I try to write the cycles of this shuffle, I get a LOT of cycles, for example: $$(2\ 3)(27 \ 2)(26\ ...
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How to show that the cycle $(2 5) = (2 3) (3 4) (4 5) (4 3) (3 2)$

I generally do not have any problem multiplying cycles, but I've seen on Wikipedia that $$(2 5) = (2 3) (3 4) (4 5) (4 3) (3 2). $$ I started following the path of $2$ on the right: ...
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Free product of two algebras and actions of algebras.

Let $A, B$ be two algebras. Suppose that $A$, $B$ acts on $V$. Then we have two maps $$ \delta_1: A \otimes V \to V, \\ \delta_2: B \otimes V \to V, $$ which satisfy the axioms of actions. Do we ...
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e solutions of gallian abstract algebra 4th edition [on hold]

teacher, Please help me, I am trying my level best but still not found ebook for solutions of Gallian abstract algebra. But I hope I will get soon. Thanks a lot
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Minimal polynomial for primitive root [on hold]

Let $p(x)$ be a polynomial over $Q[x]$. Prove that if $p(x)$ has as root a 5th primitive root of unity, then all 5th primitive roots are roots of $p(x)$ This would mean $p(x)$ is a multiple of $f(x) ...
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Partition of a group such that an operation can be defined

I'm struggling with Problem 43 of 3.1 of Dummit's algebra book. The problem is: Assume $P=\{A_i\}$ is any partition of $G$ with the property that a "quotient operation" is defined as follows: to ...
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Prove that for a group with even order $2k$, there is a subgroup $K$ with order $k$

I'm trying to understand the proof my teacher did: Consider a subgroup $H$ of $G$. If $H$ is not contained in $A_n$, then we can say that there exists at least one permutation in $H$ that is odd ...
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Extending a Field Monomorphism

In theorem A3.5 of Ash's book Abstract Algebra: The Basic Graduate Year (page 20 in this pdf), the author set out to prove the following. Let $\sigma: F \rightarrow L$ be a field monomorphism ...
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Binary solutions of multivariate polynomial system in special (factored) form.

In my personal research I've run into a system of multivariate polynomials (with coefficients in a field). I am aware that there is no polynomial time algorithm (in the number of indeterminates) for ...
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67 views

Permutations of the elements of $\mathbb Z_p$

Note Added by Robert Lewis, 2 August 2015 3:04 PM PST in an attempt to provide background, motivation, and other context for this engaging problem: This problem essentially asks for a method of ...
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2answers
25 views

Relation between a group's cardinality and number of subgroups

Why are these following situations not possible? A. An infinite group has finite number of subgroups B. An uncountable group has countable number of subgroups. Any infinite ...
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Is $K := \mathbb{Q}(\cos (2\pi / 11))$ a Galois extension over $\mathbb{Q}$?

I believe that it is because $\cos(2\pi / 11) = (\zeta + \zeta^{-1})/2$ where $\zeta = e^{2\pi i/11}$ is a primitive $11$-th root of unity, and so $K$ is a subfield of $\mathbb{Q}(\zeta)$ with ...
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What is the fastest, most correct way to solve this simultaneous of two linears?

\begin{eqnarray*} (x+2)/5-((y+2)/4) &=& 2-(x/3) \\ (x+5)/4+((x-y)/5) &=& y+5 \end{eqnarray*} What is the fastest, most correct way to solve this simultaneous of two linears?
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How to solve conjugation equations in group theory

Given the permutations $(12)(34)$ and $(56)(13)$ find $a$ such that $$a^{-1}xa = y$$ I just realized that I don't know how to solve this exercise. My book don't even give examples of how to solve ...
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Associate a discrete valuation ring to a field $k$.

I have a field $k$ of positive characteristic $p$, not necessarily perfect. Can i find a discrete valuation ring that have $k$ as residue field and field of fractions $K$ of characteristic zero? I ...
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Prove gN in G/N has infinite order.

Let G be a Group and let g in G. Let g have infinite order and suppose that N is a finite normal subgroup of G. Prove that the Element gN in G/N has infinite Order. I started by proving by ...
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30 views

Number of groups with a certain property

Let $S$ be the collection of groups ( isomorphism classes) $G$ which have the property that every element commutes with identity and itself only . Then what is the ...
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What does it means to multiply a permutation by a cycle? $\pi(x_1\cdots x_n)\pi^{-1}=(\pi(x_1)\cdots\pi(x_n))$

I have to prove that $$\pi(x_1\cdots x_n)\pi^{-1}=(\pi(x_1)\cdots\pi(x_n))$$ but I can't understand what this means. My book doesn't defines what a permutation and cycle product would be. So, for ...
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Polynomial Interpolation When $y_i$'s are Permuted

Recall, if we have a $d$-degree polynomial $f$, evaluate it at $\textbf{x}=(x_1,\ldots,x_n)$ we would get $\textbf{y}=(y_1,\ldots,y_n)$, where $f(x_i)=y_i$ and $d+1 \leq n$. The reverse is also true, ...
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Isomorphism of polynomial rings [duplicate]

I am trying to do exercise 3.6.F in Ravil Vakil's algebraic geometry notes : http://math.stanford.edu/~vakil/216blog/FOAGapr2915public.pdf We fix a field $k$. It comes down (or so I think) to proving ...
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Is every algebraic integer a sum of roots of $x^n - a$?

A complex number is said to be an algebraic integer if it is a root of a monic polynomial with integer coefficents. For example any root of the polynomial $x^n - a$ for $a \in \mathbb{Z}$ is an ...
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When $mB \neq B$? $m$ is a maximal ideal of $A$, $A \subseteq B$.

Let $A \subseteq B$ be commutative rings, $m$ a maximal ideal of $A$. When $mB \neq B$? This is true when $A \subseteq B$ is faithfully flat. (If I am not wrong, this is also true when $A \subseteq ...
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Algebraic structures without the axiom of closure

What's the name of an algebraic structure that doesn't satisfy the axiom of closure? For example, if a magma is composed of a set and a operation, which satisfies closure, what would we call the same ...
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A question on coalgebras(2)

Assume that $C$ is a coalgebra with comultiplication $\Delta:C \to C\otimes C$. The higher order comultiplication can be defined inductively as follows(with some abuse of notations we denote them by ...
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A question on coalgebras(1)

Is there a complex coalgebra $C$ with dimension at least 2 for which the scalar operators $T(x)=\lambda x$ are the only operators which satisfy $$(T\otimes T)\circ \Delta= \Delta \circ T^{2}$$ ...
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How do you evaluate elements of the Drinfeld double $D(H)$ against elements of $H^\ast$ or $H$?

Proposition 8.2 of Majid's primer on quantum groups says that if $H$ is a finite dimensional Hopf algebra with quantum double $D(H)$, then this is a factorizable Hopf algebra with quasi-triangular ...
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Why this $\sigma \pi \sigma^{-1}$ keeps apearing in my group theory book? (cycle decomposition)

I'm studying cycle decomposition in group theory. The exercises on my book keep saying things like: Find a permutation $\sigma$ such that $\sigma (123) \sigma^{-1} = (456)$ Prove that there is no ...
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Of the 16 binary operations on a two element set, which ones are commutative, associative, have an identity element, and have inverse?

If you you 16 binary operations $$(a*a)=a$$ $$(a*b)=a$$ $$(b*a)=a$$ $$(b*b)=a$$ $$(a*a)=a$$ $$(a*b)=b$$ $$(b*a)=a$$ $$(b*b)=b$$ $$(a*a)=b$$ $$(a*b)=a$$ $$(b*a)=b$$ $$(b*b)=a$$ $$(a*a)=b$$ ...
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Example of two modules M, N where the set of the $m\otimes n$ is not a submodule

If I remember my Algebra class correctly, this is in general not true because, for example, the addition isn't closed. So if I take: $B:\mathbb{R}^2 \times \mathbb{R}^2\to \mathbb{R}^{2^2}\cong ...
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Prove that there is no permutation $\gamma$ such that $\gamma (1 2) \gamma^{-1} = (1 2 3)$

I need to prove that there is no $\gamma$ such that: $$\gamma (1 2) \gamma^{-1} = (1 2 3)$$ First of all, I'll try to write $\gamma$ in a generic way: $$\gamma = (a b c) \implies \gamma^{-1} = ...
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Sizes of Quotient Rings of DVRs with Finite Residue Field

If $R$ is a discrete valuation ring (DVR) with maximal ideal $\mathfrak{m}$ such that $R/\mathfrak{m}$ is finite, then all quotient rings of $R,$ namely $R/\mathfrak{m}^n$ for $n \in \mathbb{N},$ are ...
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Example of $S[1/a] \cong S[1/b]$ as rings via $\phi$, where $S$ is a UFD, $a, b \in S$, and $\phi(U(S)) \neq U(S)$.

Above, $U(S)$ refers to the units of $S$. This problem stems from reading the paper "Translates of Polynomials", where a fact about a ring isomorphism between $S[1/a]$ and $S[1/b]$ is proved under ...
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Can two representations with different dimensions be isomorphic?

For a finite group G and two irreducible representations, with different dimensions. How would I show that they can not be isomorphic?
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Generalization of Singular locus and non-free locus to an algebra

Let $R$ be a commutative noetherian local ring with maximal ideal $\mathfrak{m}$ and $ \Lambda $ be a noetherian $ R $-algebra. Recall that: (1) The singular locus of $R$, denoted by $\mathsf{Sing} ...
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problem in meaning of symbol in commutator subgroup

i was reading paper "OUTER AUTOMORPHISMS IN NILPOTENT p-GROUPS OF CLASS 2, H. LlEBECK" in page 2 there is a symbol i dont get. if G is generated by a basis $a_λ, λ∈Λ$ and z∈Z⋂Φ(G) for σ(z,μ) be ...
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If $G$ is a field and there is an isomorphism $f\colon H/I \to G$, then does $I$ have to be a principal ideal?

I noticed that the ideal $I = \left(2, 1 + \sqrt{-7}\right)$ follows the definition of a non-principal ideal. I took two random elements from $\mathbb Z[-7]$, say $ a + b\sqrt{-7}$ and $ c + ...
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Indecomposable commutative rings [on hold]

Let $R$ be a commutative ring. Can we say that $R=\bigoplus_{i\in I}R_i$ or $R=\prod_{i\in I}R_i$ where $R_i$ are commutative ring and $I$ is an infinite set?
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A simple divisible module

Let $k$ be division ring and $V$ be a left $k$-vector space of infinite dimension. Let $E=\text{End}_k(V)$ be the ring of right linear transformations on $V$. My questions are: (1) Is $V$ a simple ...
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Are there any free or fascist boolean algebras?

A boolean algebra is an algebra with the binary operations $\wedge$, $\vee$, an unary operation $\neg$, and constants $0$, and $1$, satisfying axioms. A heyting algebra is an algebra with the binary ...
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Why do we have “another” definition for the kernel?

Why does the definition $\ker(f)=\{(a,a')\in A\times A: f(a)=f(a')\}$ exist? This definition is for any sort of algebraic system and any sort of function. But which came first... this definition or ...
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Why is the coproduct $\Delta$ of the quantum double $D(H)$ an algebra homomorphism?

If $H$ is a a finite dimensional Hopf algebra, $H^\ast\otimes H$ has a Hopf algebra structure with multiplication $$ (\phi\otimes h)(\psi\otimes g)=\sum \psi_2\phi\otimes h_2g\langle ...
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Frobenius automorphism and non-split Cartan subgroup of $\mathrm{GL}_2(\mathbb{F}_p)$

For completeness I give some definitions. Let $p$ a prime number and consider $V$ a 2-dimensional $\mathbb{F}_p$-vector space. Consider $k$ a sub-algebra of $\mathrm{End}(V)$ that is a field of $p^2$ ...
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Failure of group definition with weaker axioms

In "A First Course in Abstract Algebra", Edition 7, p.43, Fraleigh writes that It is possible to give axioms for a group $\left<G,*\right>$ that seem at first glance to be weaker, namely: ...
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Show that $(t^m-1)/(t^n-1)$ is a square if and only if $(\exists s \in \mathbb{Z})\ m=np^s$

I want to show the following lemma: Assume that the characteristic of the field $F$ is $p$ and $p>2$. Then $(t^m-1)/(t^n-1)$ is a square in $F[t, t^{-1}]$ ($F[t,t^{-1}]$: the polynomials in ...
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Polynomial Interpolation and Data Integrity

This question is about polynomial interpolation and security. Please consider a scenario where we have a polynomial $f$, one of whose roots is $a$. We evaluate it at some ...
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Composition series and its number determine a group?

By Jordan-Holder thm, it is known that every finite group has a unique composition series.(Here, unique means that there is only one kinds of such series.) And it is known also that composition ...
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Is ideal an “anti-field”?

I am comparing theorems on normal subgroup and ideal from Fraleigh's, and come to this strange intuition. I hope my conclusion does not screw up, I hope I won't get ridiculed: Theorem 15.18: $M$is ...