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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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 $k$ has residue field and field of fractions $K$ of characteristic zero? I ...
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1answer
20 views

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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1answer
24 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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32 views

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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17 views

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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15 views

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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1answer
36 views

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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29 views

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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19 views

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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13 views

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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25 views

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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2answers
110 views

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 such that $\sigma (1 2) \sigma^{-1} = (123)$ Prove that there is no permutation ...
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1answer
36 views

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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31 views

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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1answer
32 views

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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1answer
30 views

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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28 views

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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1answer
26 views

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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21 views

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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55 views

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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26 views

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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1answer
23 views

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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1answer
70 views

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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4answers
554 views

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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35 views

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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1answer
70 views

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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2answers
59 views

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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1answer
43 views

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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46 views

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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2answers
44 views

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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2answers
455 views

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 ...
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2answers
48 views

Ideals of non semi-simple group rings.

I worked for a long time on complex group rings and complex twisted group rings. In those cases the algebra is semi-simple and its structure is well understood from the decomposition to irreducible ...
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65 views

Sets with $n$ prime numbers

Let $n>2$ a natural number. We define the following sets: $$S=\{1 \leq a \leq n : (a,n)=1, a^{n-1} \not\equiv 1\pmod n\} \\ T=\{1 \leq b \leq n : (b,n)=1, b^{n-1} \equiv 1 \pmod n\}$$ ...
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2answers
41 views

Question about a passage in Fulton and Harris

So I was reading the first chapter of Fulton and Harris and they are determining the representations of $S_3$. I came along this passage and had some questions What do they mean when they say "the ...
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1answer
64 views

Factoring $x^{15}−1$ into irreducible polynomials over $\mathrm{GF}(2)$

Factorize $x^{15}−1$ into irreducible polynomials over $\mathrm{GF}(2)$ The answer is $$(x+1)(x^2+x+1)(x^4+x+1)(x^4+x^3+1)(x^4+x^3+x^2+x+1)$$ but how would I find this? Please help.
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1answer
27 views

What is the name for this $R$-module? [on hold]

If $M$ is a $R$-module such that for all $x,y$ in $R$ and $m$ in $M$ then $x.y.m=0$ ... then: how is called $M$?
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4answers
129 views

Show that $t^n-1 \mid t^m-1 \Leftrightarrow n\mid m$

I want to prove the following lemma: $t^n-1$ divides $t^m-1$ in $F[t, t^{-1}]$ if and only if $n$ divides $m$ in $\mathbb{Z}$. I have done the following: $\Leftarrow $ : $n\mid m \Rightarrow ...
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Splitting even degree polynomials

I have an octic equation (degree $8$) and a sextic equation (degree $6$) in $\Bbb Z[x]$ with very large coefficients (size several hundred bits) that I know splits into two quartics and two cubics ...
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1answer
99 views

Solve $x^3+x+3=0$ by Galois's theory

Solve with radicals the following equation $x^3+x+3=0$, using Galois Theory. I'm just starting learning this and I do not have many ideas.
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2answers
33 views

Ring homomorphism and ideal that contains the kernel [on hold]

If $f:R\rightarrow S$ is a ring homomorphism and $I$ ia an ideal of $R$ such that $ker(f) \subseteq I$ then $f^{-1}(f(I))=I$ We know that $I\subseteq f^{-1}(f(I))$ but how can I use that $ker(f) ...
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1answer
40 views

A symmetric group question. [on hold]

Determine the integers $n$ such that there is a Surjective homomorphism from the symmetric group $S_n$ to $S_{n-1}$. It is a question from Artin's book. Exercise 7.5.8
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2answers
57 views

Kernel of ring homomorphism

Let $\phi: R \to R'$ be a ring isomorphism and $I$ an ideal of $R$. Define $\phi(I)=\{\phi(i): i \in I\}$. Show that $\frac RI \cong \frac {R'}{\phi(I)}$. To use the first isomorphism theorem, ...
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1answer
45 views

For each $n >2$ find all $m \in \mathbb N$ such that there exist a group $G$ of order $m$ and a surjective homomorphism $f: A_n \rightarrow G$

For each $n >2$ find all $m \in \mathbb N$ such that there exist a group $G$ of order $m$ and a surjective homomorphism $f: A_n \rightarrow G$< I tried to use the theorem of isomorphism, and a ...
4
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1answer
36 views

A $\mathbb Z/p\mathbb Z[G]$ submodule with no complement

Let $G$ be a group acting on a set $X$ of size $n$. Suppose $G$ acts doubly transitively. If $p$ is a prime, this naturally gives a permutation representation on the vector space over $\mathbb ...
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2answers
54 views

If $G = S_5$ and $H = \{g \in G \mid g^{5} = e\}$ how could I determine and prove whether or not $H$ is a subgroup of $G$?

I think that the this group contains the 5 element cycles and the identity e but overall I'm not sure how to prove that the product of the 2 members of H is also a 5 cycle or e.
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3answers
68 views

Why does $\sqrt{6} + \sqrt{10} + \sqrt{15}$ have four conjugates?

I am having trouble understanding how algebraic number $\sqrt{6} + \sqrt{10} + \sqrt{15}$ has four conjugates. Minimal polynomial is $x^4-62 x^2-240 x-239$ according to Wolfram Alpha. Factorized: ...
14
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1answer
83 views

Is the ring of holomorphic functions on $S^1$ Noetherian?

Let $S^1={\{ z \in \Bbb{C} : |z|=1 \}}$ be the unit circle. Let $R= \mathcal{H}(S^1)$ be the ring of holomorphic functions on $S^1$, i.e. the ring of functions $f: S^1 \longrightarrow \Bbb{C}$ which ...
2
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1answer
35 views

Showing a nonabelian group of order 21 has an automorphism that is not inner.

I've seen at least 3 proofs of this on here, but most involved techniques I don't think I'm comfortable with, so I wanted to see if this one works: Since $21=3\cdot 7$, up to isomorphism there's only ...