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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Let D be a Euclidean domain and d be the associated function.Show that if a and b are associates in D then d(a)=d(b). [closed]

I'm not even getting how to begin this.Any hints are welcome. Is < a >=< b > implies d(a)=d(b)? I have proved that if a and b are associates then < a >=< b >,but i'm not getting how to ...
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2answers
54 views

Notation in commutative algebra

I am doing some exercises on commutative algebra and came along the following expressions, which were not elaborated on. Is someone familiar with them? The first is for $p$ a prime number ...
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1answer
57 views

How many roots does the polynomial $p(z) = z^8 + 3z^7 + 6z^2 + 1$ have inside the annulus $1 < |z| < 2$?

How many roots does the polynomial $p(z) = z^8 + 3z^7 + 6z^2 + 1$ have inside the annulus $1 < |z| < 2$? I know I can use Rouche's Theorem. I'm just not sure how. It states that $|f(z) − g(z)| ...
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33 views

Primitive element in multivariate Galois field [on hold]

On Singular CAS I can define a Galois field $(2^3)$ with $(x,y,z)$ variables. But I am not able to understand how $a^3+a+1$ is still its primitive element. General example taken in books is always ...
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36 views

Torsion-free divisible module over a commutative integral domain is injective. [duplicate]

This question is from the book Basic Algebra by P.M. Cohn. Show that a torsion free divisible module over a commutative integral domain is injective.
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25 views

Maximal Ideal of ring $C[0,1]$ [duplicate]

Prove that an ideal $M$ of the ring $C[0,1]$ is maximal iff there exists some $a$ in $[0,1]$ such that $M=\{f \in C[0,1]:f(a)=0\}$.
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57 views

Why are there no “continuous maps” in algebra. [closed]

Or maps that behave similarly? Sorry if this is a strange question.
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48 views

Group generated by $x , y$ is non-commutative when $x^2 \cdot y^{-3} = I$.

The problem: Suppose group $G$ with generators $x$ and $y$ is defined by the relation $x^2 \cdot y^{-3} = I$. It is necessary to show that the group is non-commutative. I failed to solve the ...
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0answers
11 views

Sum of nil right ideals as an ideal

I have two questions: 1) If $S$ is the sum of all right nil ideals of a ring $R$ (with unity), is it true that $S$ is a two-sided ideal? It is clear for me that $S$ is a right ideal (and it is nil if ...
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1answer
50 views

Is it possible to say that if a A is a set of elements of a semigroup S, there exists some other semigroups on the alphabet A?

Suppose we have a finite semigroup, its element's set is A = {1,2,...n}, can we get some other semigroups whose element's set is A too ?
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26 views

The product of dg Lie algebras

I am trying to understand what are products and coproducts in the category of dg Lie algebras. I am okay with coproducts. For products, however, this Wikipedia article says that given ...
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11 views

Orthogonal projection of skew-symmetric form

It is a question from the book Algebra by Michael Artin: 8.8.2 Let W be a subspace on which a real skew-symmetric form is nondegenerate. Find a formula for the orthogonal projection $\pi:V\to W$
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1answer
54 views

Any nonabelian group of order 12 is isomorphic to A4, D6, or Z3 X Z4 [on hold]

Can someone show me the proof for : Any nonabelian group of order 12 is isomorphic to $D_6$, $A_4$, or $\mathbb{Z}_3 \times \mathbb{Z}_4$ Ive seen a few proofs where this is included in also ...
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1answer
30 views

Show this product of principal ideals is principal

Let $R$ be a commutative ring with $1$ and $I$ an ideal. Also let $B$ be a principal ideal, and $A=\{a\in R\;|\; aB\subseteq I\}$. I want to show that if $A$ is also principal then $I$ is principal. ...
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1answer
31 views

A finite cancellative monoid is a group? Need help seeing why the following is a false proof.

I proved this in the following way is our exam, but got $0$ out of $7$ points for it, however I fail to see why its wrong: Let $a\in S$ be arbitrary. Since $S$ is finite, there has to be some $m ...
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1answer
23 views

Lifting homomorphism when module is direct summand of free module

Let $N,M,M',N'$ be modules such that $F = N \oplus N'$ is a free module. Let further $f: M \rightarrow M''$ be a surjective homomorphism. I would like to show that for any homomorphism $\phi: N ...
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2answers
27 views

Show that the union of a chain of ideals is an ideal.

Here is my proof: Let $I=I_1\cup\ I_2\cup\ I_3 \cup\ ..... \cup\ I_n$, $a\in I$ and $r\in R$. Then $a\in I_i$ for some $i$ varying from $1$ to $n$. Since $I_i$ is an ideal of $R$, we have $ar\in ...
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52 views

Determining the presentation matrix for a module

I am trying to study some module theory using the book "Algebra" by Michael Artin (2nd Edition, to be precise), and I can't really fathom what is written in Section 14.5. Left multiplication by an ...
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1answer
73 views

Is there any efficient algorithm for computing all semigroups of order n?

I found a paper, but here the author solves a bit different problem. My question is: Is there any efficient algorithm for computing all semigroups of order n?
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1answer
18 views

$N(R)$ when $R$ is a P.I. ring

The set of nilpotent elements $N(R)$ of a ring $R$ with identity is not necessarily a right ideal (or even a subgroup) as it is seen in the ring of $2×2$ matrices over $\mathbb Z$. But, my question ...
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1answer
54 views

Show that $U_{14}\cong U_{18}$?

Because $|U_{14}|=|U_{18}|$ and both of them is cyclic and commutative, so i just need define a function $f : U_{14}\to U_{14}$ that f is homomorphism and bijective. $f(1)=1, f(3)=5, f(5)=7, f(9)=11, ...
3
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1answer
32 views

On the degree of cyclotomic fields extension.

Let $F$ be a field and $n$ be any integer satisfying $(n,\text{char}(F))=1$ when $\text{char}(F)\ne 0$. Let $\xi$ be a primitive $n$-th root of unity. I know that $\text{Gal}(F(\xi)/F)$ is isomorphic ...
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2answers
43 views

Proof of a property of set of all one-to-one mappings

Let $S$ be a nonempty set and $A(S)$ be the set of all one-to-one mappings of $S$ onto itself. I.N. Herstein in Topics in Algebra says (in page 28) that whenever $S$ has three or more elements, we can ...
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1answer
40 views

Determine the degree of an extension field over $\mathbb{Q}$

Let $\alpha = e^{\frac{i\pi}{6}}$. Compute $[\mathbb{Q}(\alpha):\mathbb{Q}]$ and find the minimal polynomial of $\alpha$, $m_{\mathbb{Q}}(\alpha)$. I can see clearly that $\alpha^6+1=0$ but I ...
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2answers
56 views

Proving that $f(x)$ is irreducible in $F[x]$ iff $\phi(f(x))$ is.

This is what I'm proving: Let $F$ be a field. Let $\phi : F[x]\to F[x]$ be an isomorphism such that $\phi(a)=a$ for every $a\in F$. Prove that $f(x)$ is irreducible in $F[x]$ iff $\phi(f(x))$ is. ...
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1answer
23 views

Applying hom-functor on short exact sequence

Let $N, M, M', M''$ be $R$-modules. Given homomorphisms $f: M' \rightarrow M$ $g: M \rightarrow M''$ $\psi: N \rightarrow M$ with $\operatorname{im} f = \ker g$, $g \circ \psi = 0$ and $g$ ...
4
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1answer
41 views

Brauer group of cyclic extension of the rationals

I am trying to compute the relative Brauer group of the cyclic Galois extension $L=\mathbb Q[x]/(x^3-3x+1)$ of $\mathbb Q$. I know that $$ \mathrm{Br}(L/\mathbb Q)\cong H^2(G,L^*)\cong\mathbb ...
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1answer
24 views

Good reference to study Free Algebras

I am interested in studying free algebras as Free Pre-Lie Algebras, Free Dendriform Algebras etc. But I dont know what a free algebra is in general. I found this definition on Wikipedia but it does ...
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How do I get the Rational Canonical Form from the minimal and characteristic polynomials?

Let's say I have the minimal polynomial and characteristic polynomial of a matrix and all its invariant factor compositions. How do I get a rational canonical form matrix from this?
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2answers
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Confusion about elements in fields, like -1 in Z5

I'm learning field and ring theory, and I've repeatedly seen the usage of -1, -2 and -3 as elements of $\mathbb{Z}_5$. As far as my knowledge goes, $\mathbb{Z}_5$ consists of {0,1,2,3,4}. This is ...
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Difference between the set of generators and the alphabet of a free group

What do we mean by saying "a semigroup P is presented by generators and relations". Isn't it right only for the free semigroups? If it's right, we can't distinguish some two semigroups if they are ...
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34 views

Understanding isomorphism of Hom's

During some reading I came across the statement $$Hom_R(M,N) \otimes R/I \cong Hom_{R/I}(M/IM,N/IN) $$ where $M,N$ are $R$-modules and $I$ an ideal of the commutative ring $R$. This is proved, under ...
4
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1answer
60 views

$K/\mathbb{Q}$ contains a complex number - is complex conjugation in $\text{Aut}(K/\mathbb{Q})$?

Suppose $K$ is a finite extension of $\mathbb{Q}$. Suppose there is some complex number in $K$. Is it necessary that $\tau$, complex conjugation, is a member of $\text{Aut}(K/\mathbb{Q})$? I feel ...
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1answer
36 views

Degree of field extension over a fixed field

Suppose $L$ is a field and $H$ is a finite subgroup of $Aut(L)$. Then $[L:L^H]=|H|$ I've seen a proof of this that uses the Rank-Nullity theorem but it's rather cumbersome and difficult to remember ...
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67 views

Abstract Mathematics - Group theory and isomorphism

I have been trying to solve two problems, but I am stuck. Can anybody provide me with some links or theory to solve the following problems? The problems are from a study guide and the test exercises ...
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63 views

order $a$ = 5, $a^3b = ba^3$. show that that $ab = ba$.

Let $a, b$ be elements of a group $G$. Suppose that a has order $5$ and that $a^3b = ba^3$. I want to show that that $ab = ba$. Here is what I think: We know that we have $a^1, a^2, a^3, a^4, a^5 = ...
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2answers
35 views

Finding the parity of a permutation “exclusively”?

I'm trying to find the parity of permutations such as $(2468)$. What makes it possible to find the "exclusive" parity of such permutation? I.e. that if one tries to express $(2468)$ as a product of ...
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1answer
39 views

Commutativity of ring $R$ necessary for $\mathrm{Hom}_R(M,M')$ being an $R$-module

Why do we need $R$ to be commutative if we want $\mathrm{Hom}_R(M,M')$ (where $M$ and $M'$ are $R$-modules) to be an $R$-module itself? I tried to find out which axiom for modules does not hold if ...
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2answers
28 views

If $\sigma=(a_1 a_2 … a_n)$ and $|\sigma|$ is odd, then what is $\sigma^2$?

I'm trying to understand the way to infer the power of a permutation. If $\sigma=(a_1 a_2 ... a_n)$ is a $k$-cycle and $k=|\sigma|$ is odd, then how can I infer what $\sigma^2$ is?
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$x\in\overline{F}$ is in $F(\sqrt{F})$ $\iff$ $F\subset F(x)$ is finite Galois extension with Gal$(F(x)/F)$ abelian of exponent $2$

Let $F$ be a field of characteristic that is not $2$. I want to prove that $x\in\overline{F}$ is in $F(\sqrt{F})$ $\iff$ $F\subset F(x)$ is a finite Galois extension for which the group ...
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1answer
24 views

The question about second isomorphism theorem

If $G$ is a group and $N \trianglelefteq G$ and $K \leqslant G$, and $N_1 \leqslant N$, $N_1 \trianglelefteq G$, then can we say $NK/N_1 \approxeq K/K \cap N_1$?
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25 views

Maximum length of product of disjoint cycles?

This proof concerning the largest possible order of a permutation in $S_{10}$ uses some theorem for inferring that in $S_{10}$ the length $k_1+...+k_n$ of a product of disjoint permutations $k_1, ..., ...
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9 views

Submodules of direct sum of simple modules [duplicate]

Is it true that if $N$ is a simple module, then the only non trivial submodule of $N\oplus N$ is $N$? I am aware of the "diagonal" submodule $\{(x,x)\mid x\in N\}$, but that is isomorphic to $N$?
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31 views

Extension Operator.

I am working on my thesis about completion and extensions from an algebraic point of view. We have the closure operator which takes subsets to subsets with 3 criterias to meet $X\subseteq C(X)$ ...
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1answer
36 views

Are the free monoids always infinite?

It's the Wikipedia's definition of the free monoid: **In abstract algebra, the free monoid on a set is the monoid whose elements are all the finite sequences (or strings) of zero or more elements from ...
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How to proof equivalent condition of algebra morphism and coalgebra morphism about Hopf algebra

My question is why $\mu$ is multiplication preserving iff $\mu\circ m=(m\otimes m)\circ(1\otimes \tau \otimes1)\circ(\mu \otimes \mu)$ and this holds iff $m$ is comultiplication preserving, where ...
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81 views

Why is the Galois group of the polynomial $f(x)=x^5-x-1$ isomorphic to $S_5$?

Currently I am trying to understand why quintic (and higher degree) polynomials are not soluble by radicals, and one of the easiest examples of a quintic polynomial which has a nonsoluble Galois ...
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22 views

inverse of a root of f in an extension field K. [closed]

Let $f(x) = x^n + a_{n−1}x^ {n−1} + \cdots + a_0$ be an irreducible polynomial over $F$, and let $\alpha$ be a root of $f(x)$ in an extension field $K$. Determine the element $\alpha^{-1}$ explicitly ...
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39 views

Principal Ideal using coordinates?

I thought I understood principal ideals but now im stuck... I want to find the elements of the principal ideal $\langle(1,0)\rangle$ in the ring $\mathbb Z_3\times \mathbb Z_3$ with $+_3$ and $*_3$ in ...