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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Does $a^2(x) = b^2(x) (1 - x^2)$ imply $a = b = 0$?

I have been working through an exercise and I have found out that $a^2(x) = b^2(x) (1 - x^2)$, where $a(x), b(x) \in \mathbb{R}[x]$. Is this enough to deduce $a = b = 0$? I think so, because ...
0
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3answers
46 views

Proof of ring isomorphism

Proof that $Z[X]/(X^2-22) ≈ Z[\sqrt{22}]$. I have tried all sorts of things to resolve this but I don't know how to wrap my head around it. Can you please explain how to solve these kind of ...
-1
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1answer
98 views

Number of Subfields of the field [duplicate]

The Number of subfields of a field of Cardinality $2^{100}$ is a) $2$ b) $4$ c) $9$ d) $100$ The Possible Number of Divisors of 100 might be the answer, hence 9 is the answer.. ...
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2answers
46 views

Are these rings fields?

Are the following rings fields? 1) $\Bbb Q[x] /\langle x^2+1\rangle$ Since a polynomial ring taking values on any field is a E.D, and hence a P.I.D, this is a field iff the ideal is prime or ...
1
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1answer
28 views

Quotient ring is a ring homomorphism

Why is this a ring homomorphism? $$\phi:R\to R/I$$ where $I$ is an ideal, given by $\phi:r\mapsto r+I$. To be a ring homomorphism it needs to be a homomorphism of addition and multiplication, i.e: ...
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0answers
36 views

Number of groups of a certain order given: 1) finitely generated abelian, 2) subgroups, 3) not necessarily finitely generated

1) How many finite abelian groups are there of order $1000$? Well via the fundemental theorem of finitely generated abelian groups, we look at the factorisation for $1000$. $1000=2^3*5^3$ and there ...
3
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1answer
52 views

Are these subrings of $\Bbb Q$?

Are the following subrings of $\Bbb Q$? 1) The set of non-negative rational numbers. No since we don't have any additive inverses, and the subring should be armed with an Abelian group for ...
0
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3answers
31 views

Prove for every $a$ in $I$ and every $b$ in $J$ that $ab=0$.

$I$ and $J$ are respectively right and left ideals of ring $R$. $I$ and $J$ have no elements in common other than $0$. Prove for every $a$ in $I$ and every $b$ in $J$ that $ab=0$. I have ...
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2answers
56 views

In general, how many non-abelian groups of order $p^3$?

In general, how many non-abelian groups of order $p^3$? For $p=2$, there are 2 groups namely $Q_8$ and $D_4$. For $p=3$, there are again 2 groups. Can we ...
4
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2answers
42 views

Prove that: $B/A \triangleleft R/A$ if $A \subseteq B ;\ \ A, B \triangleleft R $(ring)

Prove that: $B/A \triangleleft R/A$ if $A \subseteq B ;\ \ A, B \triangleleft R $(ring) : $ \triangleleft $ means ideal. I need this proof to continue on, I'm told it's not that hard, but I just ...
2
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1answer
25 views

Finding the subfields of the cyclotomic field of order $5$

This is part of an exercise from Hungerford's Algebra: Find all intermediate fields in the field extension $F_5/\mathbb{Q}$, where $F_5$ is the cyclotomic extension of $\mathbb{Q}$ of order $5$. ...
1
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2answers
72 views

Converse of Hilbert's Nullstellensatz [duplicate]

I'm referring to the following version of the Nullstellensatz: If $k$ is algebraically closed, then every maximal ideal of $k[X_1,\ldots,X_n]$ is of the form $(X_1-\lambda_1,\ldots,X_n-\lambda_n)$ ...
1
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1answer
32 views

Equations in local rings

Let $R$ be a finite commutative local ring with identity. Assume that every ideal in $R$ is principal. Let $u$ and $v$ be units in $R$ and let $z\neq 0$ be a zero divisor. I think that $uz=vz$ ...
0
votes
1answer
47 views

Is finding generators of finite fields hard?

Task: Given $n$, find a generator of $GF(n)^*$. Is there any evidence this is hard? Maybe a reduction from another problem presumed hard? Finding the orders of elements should be hard because I ...
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0answers
45 views

Determine all maximal and prime ideals of the polynomial ring $\Bbb C[x]$

Determine all maximal and prime ideals of the polynomial ring $\Bbb C[x]$ My attempt: Note that $\Bbb F[x]$ where $\Bbb F$ is any field is a Euclidean domain, and importantly, that means that ...
4
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3answers
32 views

Is the set of all rational numbers with odd denominators a subring of $\Bbb Q$?

Is the set of all rational numbers with odd denominators a subring of $\Bbb Q$?(When the fraction is completely reduced) I have tried to apply the subring test on this, and this means I want to show ...
2
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3answers
45 views

If $H$ and $K$ are subgroups of $G$ whose orders are coprime. Is $H\cap K$ a subgroup of $H$ and $K$?

If $H$ and $K$ are subgroups of $G$ whose orders are coprime. Is $H\cap K$ a subgroup of $H$ and $K$? They both have the same identity, so we know at minimum we have $\{e\}$ so it is the trivial ...
2
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0answers
39 views

Can my students ask questions in Chinese here, and can I write answers in Chinese? [migrated]

我想要讓我的抽象代數班級的學生能夠透過網路向我詢問問題, 我可以建置一個私人的社群, 並且讓他們可以在這裡用中文問問題, 並且讓我用中文回答他們嗎? Google translate produces: I want to let my abstract algebra class of students through the Internet to be able to ask ...
0
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1answer
29 views

How to use class equation for determining the center of $S_4$

How to use class equation for determining the center of $S_4$ $$|G|=|Z(G)|+\sum_x [G:C_G(x)]$$ So I guess I need to find $$|G|-\sum_x [G:C_G(x)]=|Z(G)|$$ Well $|S_4|=4!=24$ and $C_G(x)$ is the set ...
5
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2answers
101 views

Existence of ring homomorphism from $\mathbb{Z}[\frac{i}{2}]$ to a finite field

Does a ring homomorphism from $\mathbb{Z}[\frac{i}{2}]$ to a finite field with characteristic $p\equiv 3 \bmod 4$ such that the unity is mapped onto the unity exist? Thank for your help.
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2answers
52 views

Trying to understand a proof about onto/1-1 mappings (from Herstein's Topics in Algebra)

I am working on some problems in a book I have and I want to make sure that I have an accurate possible proof. That is, I want to make sure I actually understand/ can justify the reasoning. (some of ...
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0answers
63 views

In the ring of polynomials $F[t]$, every ideal is principal

Let $$\langle a \rangle=\bigcap_{ a \in I} I$$ $$ \langle a \rangle =\{ Ira + na, r\in R,n \in Z\}$$ What is unclear is why$$ I=\langle 0 \rangle$$ proves that I is a principal ideal. My definition ...
3
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1answer
19 views

Abstract algebra. Proof of: Let $F$ be a finite field and $P$ an irreducible polynomial upon $F$. Then $(F[t]|_{\equiv_P}, + , \ast)$ is a field.

Division of polynomials I put what is unclear to me in between three asterisks bounding the unclear lines... $\equiv_{P}$ is defined as ($\forall Q, S \in F[t]$) $Q\equiv_{P} S \iff ...
3
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3answers
36 views

Abstract Algebra, a question on polynomials .

Theory : Every polynomial $P_n$ , of n-th power over the field $R$- real numbers, can be written as a product of polynomials of the power $<3$, who's coefficients are real numbers. Proof: ...
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votes
3answers
81 views

Factor the polynomial [duplicate]

Factor the polynomial $X^3-X+1$ in $F_{23}$ and $X^3+X+1$ in $F_{31}$. How can I know in which way to factor a polynomial mod $p$? Is there some specified method to do that? Thanks.
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2answers
21 views

Proving that an $FG$-homomorphism is surjective

Assume that $V$ is an $FG$-module.Prove that the subset $$V_0 = \{v \in V : vg = v \space \forall \space g \in G \}$$ is an $FG$- submodule of $V$. Also show that the function $$\phi: v \to ...
2
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1answer
89 views

Ring of integers of a cyclotomic number field

Let $\omega$ be the primitive $n^{th}$ root of unity. Consider the number field $\mathbb Q(\omega)$. How to show that the ring of integers for this field is $\mathbb Z(\omega)$? Also, find the ...
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1answer
19 views

Abstract algebra, polynomials , division.

It says here that $S,T$ are polynomials and $S=0, T \neq 0$ then $GCD(S,T)=a^{-1}T$ where $a$is the leading coefficient of polynomial $T$. Why is this?
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1answer
48 views

Splitting fields, eliminating zeros

Is splitting field $\mathbb{Q}(\sqrt{2+i \sqrt{6}}, \sqrt{2 - i \sqrt{6}})$ equal to $\mathbb{Q}(\sqrt{2+i \sqrt{6}}, i\sqrt{6})$? I mean, can we eliminate $\sqrt{2 - i \sqrt{6}}$ like this?
2
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2answers
145 views

Why is the observation that proof of the Fundamental Theorem of Algebra requires some topology not tautological?

I have heard it mentioned as an interesting fact that the Fundamental Theorem of Algebra cannot be proven without some results from topology. I think I first heard this in my middle school math ...
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0answers
21 views

Splitting fields and zeros

If we have splitting field $\mathbb{Q}(x_1,x_2,x_3,x_4)$ where $x_1=-x_2$ and $x_3=-x_4$ we can rewrite it like $\mathbb{Q}(x_1,x_3)$ and then we can multiply $a=x_1x_3$ and rewrite it like ...
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40 views

A question about the dual map of $A \otimes B \to B$.

Let $A$, $B$ be two algebras. Suppose that we have an action $\varphi: A \otimes B \to B$ and there is a pairing $\psi: A \otimes B \to \mathbb{C}$. The action $\varphi$ induces a map $\delta: B \to ...
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votes
2answers
41 views

Is every epimorphism $M\longrightarrow A$ a retraction? [closed]

Let $M$ be a $A$-module (where $A$ is an algebra over a commutative ring $R$). How can I show: $(i)$ Every epimorphism $M\longrightarrow A$ is a retraction; $(ii)$ There are monomorphisms ...
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1answer
18 views

The generators of the group $\langle\mathbb{Z}_n,\oplus\rangle$ are all $g \in \mathbb{Z}_n $ for which $\gcd(g,n)=1$ [duplicate]

I'm trying to find a proof of this: The group $\langle\mathbb{Z}_n,\oplus\rangle$ is cyclic for every $n$, where $1$ is a generator. The generators of the group ...
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1answer
26 views

Product-preserving functors

Let $\bf{S}^0$ be the dual category of sets, let $\mathcal{U}_{\infty}$ be a category and $A_{\infty}:\bf{S}^0\longrightarrow\mathcal{U}_{\infty}$ be a functor which is bijective on the object ...
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0answers
53 views

How (possibly) many subfields does $\mathbb{F}$ has that extend $\mathbb{Q}$?

$\mathbb{F}/\mathbb{Q}$ is a finite extension. Denote $\dim_{\mathbb{Q}}\mathbb{F}=n$. How (possibly) many subfields does $\mathbb{F}$ has that extend $\mathbb{Q}$? Thoughts: if ...
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2answers
38 views

Let $G$ be a group of order 45. Properties of subgroups of $G$. [on hold]

Let $G$ be a group of order 45. Then which of the following is/are true? $G$ has an element of order 9. $G$ has a subgroup of order 9. $G$ has a normal subgroup of order 9. $G$ has a normal subgroup ...
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1answer
55 views

How many real roots for this cubic? [closed]

How many real roots of the equation $(x-a)^3+(x-b)^3+(x-c)^3=0$ are there?
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20 views

Construction of algebraic closure

I am recently studying abstract algebra and I found a great difficulty on understanding algebraic closure. By definition, an algebraic closure $\bar F$ of $F$ is an algebraic extension of $F$ which is ...
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27 views

Product of disjoint cycles and product of transpositions

$\alpha=(3412)(245)\in S_5$ and I have to 1) write this as a product of disjoint cycles, 2) write this as a product of transpositions. 1) I can do thing by following where the elements go in the two ...
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0answers
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'Randomness' of inverses of $(\mathbb{Z}/p \mathbb{Z})^\times$

Suppose you are given the group $(\mathbb{Z} / p \mathbb{Z})^{\times}$, where $p$ is prime. Let $A_p$ denote the sequence whose $j$th element is the inverse of $[j]$. For instance, if $p = 7$, the ...
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1answer
20 views

Question about required rigour in mappings proof

I was just working on some intro problems from an algebra textbook, and one of the proofs I had seemed to make sense to me, but when I compared it to a solution given online, it was seemingly very ...
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1answer
49 views

Is the finite union of algebraic curves an algebraic curve? [closed]

Is the finite union of algebraic curves an algebraic curve? I'm kind of new to the study of algebraic curves and I believe this is intuitive.
2
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1answer
34 views

Is it true that for every $n\in\mathbb{N}$ there exists an algebraic curve $C$ and a point $p$ in that curve whose tangent plane has dimension $n$?

The title is very self-explanatory, I was thinking of finding a curve $k^n$ (k is the field with a non-singular point p, such that its tangent plane is V(0) and thus would be $k^{n}$ whose dimension ...
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0answers
57 views

Some intuition about modular multiplicative inverses

I'm trying to gain some intuition about modular multiplicative inverses. Suppose I have a prime number $n$ and the group $(\mathbb{Z} / n \mathbb{Z})^\times$, and I pick some constant $\alpha$ much ...
2
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1answer
50 views

Quotients of the ring of Laurent polynomials in one variable

I am trying to understand quotients of the ring $R=k[X,X^{-1}]$, where $k$ is a finite field. I note that $R$ is a PID since it is the localization of a PID; namely $k[X]$ localized at ...
3
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1answer
30 views

Why do roots span dual space of maximal toral subalgebra?

Suppose $\Phi$ is the root system of a semi simple Lie algebra with maximal toral subalgebra $H$. I read that $\Phi$ spans $H^\ast$. The Killing form on $H$ is nondegenerate, so $H\cong H^\ast$ by ...
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2answers
48 views

Guarantee that the intersection of a family of sets satisfies $P$ when every member satisfies $P$

1) Suppose we are given a family of sets $\{A_\alpha\}_{\alpha\in\Lambda}$ indexed by a set $\Lambda$ of arbitrary cardinal. Suppose that in addition the sets $A_\alpha$ are endowed with some property ...
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36 views

is $Hom(P,N \otimes_{End(P)} P) = N$?

This is probably well known to people who work with algebras but I couldn't find a reference. Say I have a ring A and a module P and I take B = End(P), the endomorphism ring. Let N be a B-module, is ...
1
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1answer
21 views

Relevance scoring mechanism for multiple parameters

I have a program which build few attributes those decide relevance between two objects. attributes are $a_1, a_2, a_3$ Now what are different weighing or scoring mechanism to accumulate all three ...