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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3
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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 ...
0
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1answer
14 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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0answers
20 views

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 ...
2
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1answer
20 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 ...
2
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2answers
40 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 ...
0
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2answers
25 views

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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0answers
21 views

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 ...
0
votes
1answer
39 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 ...
0
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1answer
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 ...
4
votes
2answers
103 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 ...
0
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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 ...
0
votes
1answer
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 ...
0
votes
3answers
94 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 ...
2
votes
1answer
35 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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1answer
33 views

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 $K$?
3
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0answers
25 views

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 ...
9
votes
7answers
88 views

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 $\mathbb{Z}[X]$. ...
5
votes
1answer
53 views

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 ...
0
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1answer
21 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 ...
0
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0answers
14 views

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.
2
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1answer
22 views

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 ...
0
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1answer
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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4answers
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
26 views

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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0answers
13 views

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 $L$ ...
0
votes
2answers
31 views

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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0answers
13 views

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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2answers
51 views

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 ...
0
votes
1answer
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
26 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 ...
7
votes
1answer
46 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 ...
2
votes
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 ...
0
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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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votes
2answers
37 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
votes
1answer
44 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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votes
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 ...
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vote
2answers
127 views

How to find the irreducible polynomial?

It is giving me a lot of trouble, and I'm beginning to think it's not possible. Find $\operatorname{irr}(2\sqrt{2} + \sqrt{7})$. I start like this: $x = 2\sqrt{2} + \sqrt{7}$ I have squared ...
3
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1answer
54 views

Necessarily a homeomorphism?

Let $D$ be the projective curve defined by $y^2z = x^3.$ Consider the map $f: \mathbb{P}_1 \to D$ defined by$$f[s, t] = [s^2t, s^3, t^3].$$Is it necessarily a homeomorphism? Any help would be greatly ...
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0answers
56 views

Irreducibility of $X(X-3)(X-\alpha)(X-\beta) + 1$

I'm trying to solve the following exercise: Show that for $\alpha,\beta\geq 3$, the polynomial $X(X-3)(X-\alpha)(X-\beta) + 1\in\mathbb Z[X]$ is irreducible. It is straightforward to check that ...
3
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1answer
53 views

Fields of characteristic $p$ that are isomorphic as vector spaces, but not as fields

Give an example of a field $\mathbb{K}$ of characteristic $p > 0$ and elements $a,b \in \bar{\mathbb{K}}$, such that $\mathbb{K}(a)$ and $\mathbb{K}(b)$ are isomorphic as vector spaces over ...
5
votes
3answers
66 views

Find the Galois group of the polynomial $x^{4} + x - 1$ over the finite field $\mathbb{F}_{3}$

Find the Galois group of the polynomial $x^{4} + x - 1$ over the finite field $\mathbb{F}_{3}$. I'm particurly struggling with what would be the approach to finding the splitting field of this ...
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0answers
17 views

Proving existence of unique maximal subfields of Galois extensions with particular properties

A question I am working on asks the following: Let $K / \mathbb{Q}$ be a Galois extension. Prove that there exists a unique maximal subfield $F$ of $K$ such that $F / \mathbb{Q}$ is Galois with ...
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0answers
42 views

The degree-genus formula cannot be applied to singular curves in $\mathbb{P}_2$?

(The degree-genus formula) The Euler number $\chi$ and genus $g$ of a nonsingular projective curve of degree $d$ in $\mathbb{P}_2$ are given by$$\chi = d(3-d)$$and$$g = {1\over2}(d-1)(d-2).$$ My ...
2
votes
2answers
196 views

Can Zorn's Lemma be 'inverted' like this:?

Let $R$ be a (commutative) ring not equal to $0$. I want to show that the set of prime ideals of $R$ has a minimal element w.r.t. inclusion. This may be a wholeheartedly wrong attempt, but I thought ...
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vote
3answers
32 views

How to solve a congruence using Fermat's Theorem?

I'm reading Fraleigh's A First Course in Abstract Algebra and I'm trying to understand an example (later I have to solve several problems of the same type). Little Theorem of Fermat: If $a\in ...
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0answers
46 views

A morphism which is not a comorphism of a regular map

In the lecture, we dealt with morphisms, comorphisms and regular maps. The professor then brought the following example: Let $U$ and $V$ be quasi-affine sets over $\mathbb{C}$ and let $\psi \colon ...
5
votes
2answers
52 views

The number of $3\times 3$ nilpotent matrices over $\mathbb{F}_q$ using the Orbit-Stabilizer theorem

The Fine-Herstein theorem says that the number of of nilpotent $n\times n$ matrices over $\mathbb{F}_q$ is $q^{n^2-n}$. I am trying to verify this for the cases $n=3$ using the orbit-stabilizer ...
4
votes
0answers
36 views

extending isomorphism to an automorphism

Lets say $K/F$ is a field extension and $\alpha ,\alpha '\in K$ are two distinct roots of the same irreducible polynomial in $F[x]$. there exists an isomorphism $$\psi:F(\alpha)\rightarrow ...
2
votes
1answer
22 views

References for loop rings

I recently saw a paper on alternative loop rings, as always, I am interested in all kinds of rings, and new kind of ring looked attractive. I would like to read loop and then loop rings in detail from ...