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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Proving that disjoint unions of representations are coproducts of groups

I'm working through Aluffi's Algrebra: Chapter 0 and I need some assistance with an excercise. Aluffi, Ex. II.8.7 Let $(A|R)$, resp. $(A'|R')$, be a presentation for a group $G$ in Grp, resp. ...
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$\operatorname{Rad}(k)=\operatorname{Rad}(L)$

Given a Lie Algebra $L$ on a field $F$, we define the radical of $L$ $\operatorname{Rad}(L)$ as the largest solvable ideal of $L$. We define the adjoint representation ...
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Classifying groups of order 18

I am trying to classify groups of order 18. So far, I have shown that a group $G$ of order 18 is given by $G\cong C_9 \rtimes_{\varphi} C_2$ or $G\cong (C_3 \times C_3)\rtimes_{\varphi} C_2$. If ...
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Prove that $f=x^6+ax+5$ is reducible over $\mathbb{Z_7},\forall a\in\mathbb{Z_7}$

We have $f=x^6+ax+5\in\mathbb{Z_7}$ and we have to show that it is reducible on $\mathbb{Z_7}$, $\forall a\in\mathbb{Z_7}$. Here are all my steps: For $a=0$ we'll get $f=x^6+5\in\mathbb{Z_7}$. But ...
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Fraction modulo integer in sage

I'm working on a sage script right now, I have some polynomials coefficients that are rational, and I want to apply a congruence on these coefficientss, for example: $p = 1 + (7/2)x$ the function ...
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15 views

lattices and torsion free sublatices.

I have the following statement that I cant proof, which according to my book is trivial. Let $N$ be a lattice. Let $N_1 \subset N$ be a sublattice such that $N/N_1$ is torsion free. Then it followes ...
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UTMn[D] is artinian

Why is the upper triangular matrices over a division ring D is artinian? I tried to find properties of this class of rings. The only thing I found that the jacobson radical of this ring is the ...
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1answer
45 views

To find the no. of elements of order $7$ in a field of 8 elements

Let $F$ be a field of $8$ elements and $A= \{x\in F \,|\, x^7=1 \text{ and } x^k\neq 1 \text{ for all natural number $k<7$}\}$. Then the number of elements in $A$ is: a) 1 b) 2 c) 3 d) 6 ...
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1answer
21 views

Free module over a set

I think I understand the definition of a free $R$-module over a set $X$: it is given by the set of all maps from $X$ to $R$ which vanish at all but finitely many points of $X$. The module operations ...
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Matrix ring $M_2(\mathbb{C})$, $\mathbb{C}^2$ with $M_2(\mathbb{C})$-module structure.

Let $R$ be the $2 \times 2$ matrix ring $M_2(\mathbb{C})$. let $M = \mathbb{C}^2$ with its natural $R$-module structure (just given by the usual action of $2 \times 2$ matrices on $2$-dimensional ...
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16 views

The coproduct of a family of objects of a Preorder (seen as a category)

If the coproduct of a family of objects of a Poset (seen as a category) is the least upper bound, who is the coproduct of a family of objects of a Preorder (seen as a category)? My intuition ...
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1answer
19 views

Perfect pairing induces isomorphism of tensor products

Let $M, N$ be $R$-modules and $(\cdot, \cdot): M \times N \to R$ be a perfect pairing. Wikipedia sais that this means that the map $\varphi: M \to \text{Hom}_R(N, R), m \mapsto (n \mapsto (m, n))$ is ...
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2answers
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Cyclic projective module

Let $R$ be an integral domain. If $M$ is a cyclic $R$-module which is also projective, then must there necessarily be an isomorphism of $R$-modules $M \cong R$?
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121 views

Showing that $7+\sqrt[3]{2}$ is an algebraic number

How do I go about showing that $7+\sqrt[3]{2}$ is an algebraic number? I need to show that it is the root of an integer valued formal polynomial? How do I solve these problems in general? I haven't a ...
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1answer
17 views

Prove that if $A$ is $R$-projective and $C$ is $S$-injective then $\operatorname{Hom}_R(A,C)$ is $S$-injective

In the situation $(_RA,_RC_S)$, prove that if $A$ is $R$-projective and $C$ is $S$-injective then $\operatorname{Hom}_R(A,C)$ is $S$-injective. I appreciate your help.
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Is $\oplus_{i \in I} M_i$ necessarily projective?

If $\{M_i\}$ is a collection of projective modules over a ring $R$, then must the direct sum $\oplus_{i \in I} M_i$ necessarily be projective? I understand that each direct sum of two modules is ...
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24 views

If $A \times B$ is the direct product of two rings, must $A$ necessarily be a projective $A \times B$-module?

If $A \times B$ is the direct product of two rings, must $A$ necessarily be a projective $A \times B$-module? I understand I am supposed to think of $A$ as an $A \times B$-module by identifying ...
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2answers
42 views

Proof that $a\mid x, b\mid x, \gcd(a,b)=1 \implies (ab)\mid x$

I need to prove that: $$a\mid x, b\mid x, \gcd(a,b)=1 \implies (ab)\mid x$$ What I thought was: $$a\mid x \implies x = aq_1\\b\mid x\implies x = bq_1$$ Also, since $\gcd(a,b) = 1$, we have that ...
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Show that $\mathbb{F}_q$ is a Splitting Field for Polynomial $x^q-x$

I have been given a homework problem which asks me to assume that $\mathbb{F}_q$ is a field of order $q$, and to show that $\mathbb{F}_q$ is a splitting field for the polynomial ...
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Show that $I + J = \{a + b\vert a \in I, b \in J\}$ is an ideal of R.

Let $I, J$ be ideals of a ring $R$. Show that $I + J = \{a + b\vert a \in I, b \in J\}$ is an ideal of R. Because $I,J$ are ideals of $R$, so $I,J$ both have $0$, thus $0+0=0\in I+J$. This shows ...
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41 views

What kind of algebraic structure is this?

Suppose that over a set are defined two binary operations - "+" and "*", where the first is associative and commutative, and the following law holds: $(x + y) * z = x + (y * z)$ This law is stronger ...
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39 views

Are Zero Degree polynomials Considered monics?

DO zero degree polynomials that is constant polynomials considered monic polynomials? Example F(x)=16 Does it Matter the Field or the Integral region where i take the coeficients from?Sorry if the ...
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106 views

Basic question about finite fields and characteristic

I am reading Herstein's "Topics in Algebra" and I've encountered with the following problem: If $D$ is an integral domain and $D$ is of finite characteristic, prove that the characteristic of $D$ is ...
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1answer
19 views

If $\phi(F)\neq \{0\}$, then $F\cong R$.

Let $F$ be a field and $R$ be a ring. Suppose $\phi:F\rightarrow R$ is a ring homomorphism. Show that if $\phi(F)\neq \{0\}$, then $F\cong R$. Suppose $R$ is a ring and $\phi: F\rightarrow R$ is a ...
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39 views

Squares in $\mathbb Z_p$

Let $p\neq 2$, then I want to understand that every element in $\mathbb Z_p$ (p-adic integers) is a square. For the prove one must see that $2$ is invertible in $\mathbb Z_p$. But $2$ is the element ...
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3answers
97 views

What is the intuition behind the definition of the kernel of a homomorphism

I was starting to study some algebra (groups and homomorphisms in particular) and came across the definition of the kernel (for a group-homomorphism $f:G \rightarrow G'$): $$\ker(f) = \{ x \in G \mid ...
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36 views

Why $\zeta^m \alpha \in K[\zeta]$?

In the following lemma from "The Algebraic structures of group rings" : by D.S. Passman, What does $K[G]$ contained isomorphically between $K[\zeta_1, \ldots \zeta_n] $ and $K(\zeta_1, \ldots ...
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Prove that $L=K[x]/\langle m(x)\rangle$ is an algebraic extension of $K$ [on hold]

Let $m(x)$ be an irreducible polynomial of degree $n$. How can I prove that $$L=\frac{K[x]}{\langle m(x)\rangle}$$ is: A vector space of dimension $n$, An algebraic extension of $K$?
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Does every irreducible projective cubic curve have a nonsingular point of inflection?

Does every irreducible projective cubic curve necessarily have a nonsingular point of inflection? I've been trying to construct counterexamples, to no avail, which leads me to believe the question can ...
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Why is $\mathbb{Z}[X]$ not a euclidean domain? What goes wrong with the degree function?

I know that $K[X]$ is a Euclidean domain but $\mathbb{Z}[X]$ is not. I understand this, if I consider the ideal $\langle X,2 \rangle$ which is a principal ideal of $K[X]$ but not of $\mathbb{Z}[X]$. ...
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examples of non-abstract rings?

In group theory, it helped me a lot to use symmetry groups of geometrical objects like a triangle to understand the more abstract concepts. Are there rings with a similar low level of abstractness? I ...
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22 views

Polylogarithms and the shuffle algebra

$1)$ Write $\text{Li}_2(1-\frac{1}{x})$ in terms of $\text{Li}_2(x)$ and logarithms by considering its integral representation and suitable changes of variables. Attempt: The di-log is defined as ...
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I.N.Herstein Topics in algebra problem no 2.5.18 [duplicate]

If $H$ is a subgroup of $G$.Let $N= \cap\,\, xHx^{-1} \;\;\;\forall x\in G$. Prove that $N$ is a subgroup such that $aNa^{-1}=N$. I've proved the subgroup part but couldn't show the second part.
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Localising polynomial ring $R[t]$, then for a non-maximal prime ideal $Q$, $(Q \cap R)S = Q$.

I'm trying to work out the following past paper question and I've got stuck. $R$ is an integral domain and $S = R[t]$, the polynomial ring in one variable over $R$. We have that $Q$ is a prime ideal ...
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32 views

If you have a field isomorphism and the domain is algebraically closed then so is the image?

I know it makes sense because if they are isomorphic they are practically the same thing, but what would a proof look like?
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104 views

Show that $P(x,y)=x^6+y^6$ is reducible over $\mathbb{R}$ [on hold]

Show that $P(x,y)=x^6+y^6$ is reducible over $\mathbb{R}$ In general , How do you show that a given polynomial is reducible over some field ?
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1answer
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The pushout of an epimorphism is an epimorphism

I'm reading the book "Handbook of Categorical Algebra: Volume 1, Basic Category Theory" by Francis Borceux, and in page 52 he states that "..."the pullback of a monomorphism is a monomorphism". ...
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A Lie Algebra $L$ is reductive iff it is completely reducibile as an $\operatorname{ad}_L(L)$-module

Given a Lie Algebra $L$ we say it is reductive if $\operatorname{Rad}L=Z(L)$. How can we prove that $L$ is reductive iff it is an $\operatorname{ad}_L(L)$-module completely reducibile? Suppose $L$ ...
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A basic isomorphism of modules (useful for Corollary 2.7 of Atiyah MacDonald).

Let $M$ an $A$-module, $N$ a submodule of $M$, $\mathfrak{a} \subseteq A$ an ideal such that $M = \mathfrak{a}M + N$. Then $\mathfrak{a}(M/N) = (\mathfrak{a}M+N)/N$ I am having troubles in ...
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Semi projective modules [on hold]

Consider the quotient field K of a discrete valuation ring R which is not complete. Is R-module M = K^2 is quasi-principally projective (Semi projective).Also M is direct-supplemented and amply ...
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Proof for quotient polynomial rings equivalent to field extension

I am predominantely looking for a proof, I have seen in my books and around but seem to have a hard time finding that if we let $\alpha_1,\alpha_2,...,\alpha_n$ be the roots of the minimal polynomial ...
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50 views

about maximal ideals

I don`t understand “if $x∉J$ then $J ⊂ x+J$. Please explain me and show me that every element of $J$ is in $x+J$. $J$ be a maximal ideal and suppose $xy$ is in $J$. We want to show either $x$ is in ...
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1answer
28 views

Quadratic Extensions

I am having a hard time understanding the concept of quadratic extensions. My book explains it: If the minimum polynomial of $a$ over a field $F$ has degree 2, we call $F(a)$ a quadratic ...
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1answer
47 views

Irreducible projective cubic, exists continuous surjection?

Let $C$ be an irreducible projective cubic in $\mathbb{P}_2$ with a singular point $p$. So consider $f: \mathbb{P}_1 \to C$ defined as follows. Identify $\mathbb{P}_1$ with the set of lines in ...
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3answers
70 views

If $a_1,a_2,a_3$ are roots $x^3+7x^2-8x+3,$ find the polynomial with roots $a_1^2,a_2^2,a_3^2$ [duplicate]

If $a_1,a_2,a_3$ are the roots of the cubic $x^3+7x^2-8x +3,$ find the cubic polynomial whose roots are: $a_1^2,a_2^2,a_3^2$ and the polynomial whose roots are $\frac{1}{a_1}, \frac{1}{a_2}, ...
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2answers
18 views

What is an embedding of extensions?

I'm given a definition that I don't understand. I just want to have an understanding of it. It goes as follows. We have two Field extensions $H$ and $K$ of a field $F$ and a map $v: K \to H$. They ...
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2answers
36 views

Field Extension Question for Polynomials

I cannot seem to find the answer to this question on the internet. It is a question about field extensions for an element $a,b \neq F$ but in some extension $K$. I am wondering if $F(a,b)= \lbrace ...
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2answers
39 views

Proving each automorphism of a group $G$ fixes a normal subgroup of order $p^n$ if $p\nmid\frac{|G|}{p^n}$

I have been going through Herstein's Algebra and came across this problem: "$G$ has order $p^{n}m$ where $p$ is a prime, $p$ doesn't divide $m$. Suppose $G$ has a normal subgroup $P$ of order $p^n$. ...
5
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1answer
45 views

Generated subring and finiteness

I need some help with this question: Let $A$ be the subring of $\mathbb{Q}(i)$ generated by $\mathbb{Z}[i]$, $\frac{1}{1+2i}$ and $\frac{1}{2+3i}$. Given $n\in\mathbb{Z} \setminus \{0\}$, can we ...
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1answer
52 views

Brilliant formulaes [on hold]

Hey Brilliant mathematician, i am very honored for having your time. I need general Formulas on breaking down a number to a different and being able to derive that number back, my requirements is to ...