Abstract algebra is the study of algebraic objects. Some of the more common algebraic objects are groups, rings, fields, vector spaces, modules, and other advanced topics.

learn more… | top users | synonyms (1)

3
votes
2answers
50 views

Group theory exercise - verification?

I'm self-studying abstract algebra, and this is the first non-trivial group theory exercise I've done. Although it's a well-known result, I'd like to make sure it is correct as it took a good few ...
2
votes
2answers
29 views

Transitivity Property of Separable Extensions

I was looking for some proof for the transitivity property of separable field extensions. Although this might sound like a very well-known fact and is referred to frequently, I do not seem to find a ...
1
vote
3answers
26 views

Quadratic number fields containing primitive roots of unity

A problem from Michael Artin's Algebra (Second Edition) from Fields: Determine the quadratic number fields $\mathbb{Q}[\sqrt{d}]$ that contain a primitive $n$th root of unity, for some integer $n$. ...
0
votes
0answers
24 views

two Roots questions

Just two questions on roots... 1) Can the length of roots only be defined relatively? And does length only come about because of the dot product and cartan integers? 2) This might be a weird ...
1
vote
3answers
47 views

The number of ideals in the quotient ring $\mathbb R[x]/\langle x^2-3x+2 \rangle$ [duplicate]

Finding the number of ideals in the quotient ring $\mathbb R[x]/\langle x^2-3x+2 \rangle$. Attempt: $R[x]/\langle x^2-3x+2 \rangle = \{f(x)+\langle x^2-3x+2 \rangle~~|~~f(x) \in R[x]\}$. Since ...
1
vote
1answer
16 views

$p\in\mathbb Z$ ramified in $R\Rightarrow p|\operatorname{disc}(R)$

The one in the title is Theorem 24, page 72 in the Marcus book "Number Fields". I have a problem with a detail in the last part of the proof. We have $\mathbb Q\le K$ and $L$ is a normal extension ...
2
votes
2answers
73 views

How do elements of $\mathbb{R}(xy,x+y)$ look like?

I have problems with determining what are typical elements of such field $\mathbb{R}(xy,x+y)$ In one indeterminate it is easier as $\mathbb{R}(x)=\Bigl\{\frac{f(x)}{g(x)}, g(x)\neq 0, ...
3
votes
1answer
48 views

Clifford Algebras in Characteristic Two

A Clifford algebra $Cl(V)$ for a vector space $V$ is defined to be the quotient of the tensor algebra $T(V)$ with the ideals generated by elements of the form $v \otimes v - (v,v)1$, where $(,)$ is ...
1
vote
1answer
29 views

Suppose that $\beta$ is a zero of $f(x)=x^4+x+1$ in some field extensions of $E$ of $Z_2$.Write $f(x)$ as a product of linear factors in $E[x]$

Suppose that $\beta$ is a zero of $f(x)=x^4+x+1$ in some field extensions of $E$ of $Z_2$.Write $f(x)$ as a product of linear factors in $E[x]$ Attempt: In $\mathbb Z_2: \beta^4+\beta+1=0$ Going by ...
1
vote
1answer
46 views

Let $p$ be a prime number and let $Z=\{z\in \mathbb{C}\: z^{p^{n}}=1$ for some $n \in N\}$.

Let $p$ be a prime number and let $Z=\{z\in \mathbb{C}: z^{p^{n}}=1$ for some $n \in\Bbb N\}$. (a) Show that every proper subgroup of $Z$ is of the form $H_{k}$ for some $k$, where $H_k=\{z\in C: ...
0
votes
0answers
10 views

For what values of λ is this family free (independent), spanning and a basis of R[t]≤3

The family of polynomials $F$ = {${(λ^2 − 1)t^3 + t^2, λt^3 + t − λ, (1 − λ)t^3 + t + 1, λ}$} in $R[t]_{≤3}$ I set their sum to 0 to find the values for it to be independent. $a((λ^2 − 1)t^3 + t^2) ...
2
votes
2answers
58 views

Cardinality of Centralizer of some element

Let $G$ be a finite group. How can we show that $|G/G^{'}|\leq |C_G(x)|$ for all elements $x\in G$?
0
votes
0answers
20 views

How to find out exactly invariant factor decomposition of finitely generated abelian groups

Suppose that we defined some finitely generated abelian group $G$. Now how does one find invariant or primary decomposition of $G$? We know that decomposition exists, how do we exactly state ...
-1
votes
1answer
60 views

How does the free abelian group of $M \times N $ have $M\times N$ as basis in construction of tensor product of modules?

With M and N being R-modules, how does Z(M,N) have $M\times N$as a basis and therefore becomes a free abelian group? Consider the element n(m,0) for an element$\,m\in M $ of order n. This is zero ...
2
votes
2answers
168 views

How can I find $\mathbb Z_4$ as an extension of $\mathbb Z_2$ by $\mathbb Z_2$?

Let $H=\{1,h\}$ and $A=\{0,a\}$ be groups, and $\pi:H\rightarrow \text{Aut}(A)$ be the trivial homomorphism. I have found $FS(H,A,\pi)=\{f_0,f_1\}$ and $IFS(H,A,\pi)={f_0}$ where ...
1
vote
1answer
37 views

$M_1\otimes N$ is not submodule of $M\otimes N$ where $M_1$ is submodule of $M$

I want to know the example : $M_1\otimes N$ is not submodule of $M\otimes N$ where $M_1$ is submodule of $M$ and these are $R$-modules. In the book, if $M_1\oplus M'\neq M$ then this is possible. ...
0
votes
1answer
68 views

Canonical ring map

Let $\chi:\mathbf{Z}\rightarrow A$ be the canonical map to a ring $A$, and let $p$ be a prime ideal of $A$. Then I claim that $\chi^{-1}(p)=(\mathrm{char} \ k(p))$ where $k(p)$ is the residue field at ...
2
votes
2answers
43 views

Closure of a Fundamental Weyl Chamber

Can someone explain what a "closure" of a Fundamental Weyl Chamber means? I assume it is related to an algebraic closure, but I don't see how. In addition, how does the Weyl group act on it and why ...
1
vote
2answers
30 views

Some irreducible polynomial

Is the polynomial given by $y^2-p(x)\in C[x,y]$ with $p$, all of whose roots are distinct, an irreducible polynomial? Interesting is when $p$ has degree $3$ innit
2
votes
0answers
39 views

How to prove: $pq=\binom{k}{q}-\binom{p}{q}-\binom{k-p}{1}+1$

For any two distinct primes $p, q$ there is a unique integer $k$ such that: $$pq=\binom{k}{q}-\binom{p}{q}-\binom{k-p}{1}+1$$ Where $k$ is the smallest integer greater than $p$ that is relatively ...
1
vote
1answer
47 views

Describe the elements in $Q(\pi)$

Describe the elements in $Q(\pi)$ Attempt: $Q(\pi)$ is the smallest field which contains $Q$ and $\pi$ We know that $\nexists~ f(x) \in Q[x]$ such that $f(\pi)=0$ Hence, $Q[x]/\langle p(x) ...
0
votes
0answers
36 views

Find the subgroup of S6 generated by two elements

I need some help to find the subgroup of $S_6$ generated by two elements: $a=(1 2 3 4 5 6)$ and $b=(2 6)(3 5)$ Thanks
0
votes
3answers
33 views

How many algebras of subsets of $X$ contain exactly four elements?

Let X be a set with five elements. How many algebras of subsets of X contain exactly four subsets? Well $\emptyset, X$ must be in any algebra of subsets of $X$ so that means we have to find two more ...
1
vote
0answers
41 views

Why this equality?

I'm trying to understand this proof in Fulton's algebraic book: I understood why we can assume $C$ a closed subvariety of $\mathbb P^n$ such that $C\cap U_i\neq \emptyset$, $i=1,\ldots,n+1$ . My ...
7
votes
1answer
133 views

Does the rank of homology and cohomology groups always coincide?

Let $(C_i)_{i \in \mathbb{Z}}$ be a chain complex of free abelian groups. Does the rank of the homology and cohomology groups of $(C_i)_{i \in \mathbb{Z}}$ always coincide, i.e. is ...
1
vote
1answer
32 views

Clarifying some elementary Orbit and Stabilizer questions

I have some elementary questions in learning about groups and I just want to be sure I am on the right track. Your help is greatly appreciated. Let $A = \{ \begin{pmatrix} a & b \\ 0 & ...
9
votes
0answers
78 views

Minimal and characteristic polynomials on tensor product spaces

Given two finite-dimensional vector spaces $V$ and $W$ over a common field $k$ as well as linear transformations $\varphi \colon V \to V$ and $\psi \colon W \to W$, what (if anything) can one say in ...
1
vote
2answers
82 views

Show that $Z\times Z$ is not cyclic… [duplicate]

The full problem is as stated in the title. I am here to check if this is a valid proof. I thought it would be easiest using Linear Algebra. Recall that an infinite cyclic group is isomorphic to ...
3
votes
1answer
58 views

Proving that the intersection of a Sylow p-group with a normal subgroup is also a Sylow p-group

An exercise from Dummit and Foote pg. $101$ ex. $9$ asks to show the following: Let $G$ be a group of order $p^{a}n$ where $p$ does not divide $n$ and let $N\unlhd G$ so that $|N|=p^{b}m$ where ...
0
votes
1answer
31 views

Doubt in the definition of closed subvarieties

I'm trying to understand this definition in Fulton's algebraic curves: In order to be $Y$ a variety, $\overline Y$ has to be an irreducible algebraic set of $\mathbb ...
1
vote
1answer
36 views

killing form and the dot product

When going from talking about roots as functionals to talking about roots as vectors in a Euclidian space (root system), does the killing form become the dot product? Are the killing form and dot ...
1
vote
1answer
47 views

$\dim(R/x) = \dim(R)-1$ for Noetherian integral domains?

Let $R$ be a Noetherian integral domain of finite Krull-dimension and $0 \neq x \in R$ a non-unit. Do we have $\dim(R/x) = \dim(R) -1$ in general? If this is wrong, does it change something if we ...
-1
votes
0answers
22 views

A commutative ring that satisfies $(x=0) \rightarrow ((\forall z \forall y \,\, x = yzz) \vee (\forall y \forall z \,\, x=yz+zy))$

Can anyone give an example of a commutative ring that satisfies $(x=0) \rightarrow ((\forall z \forall y \,\, x = yzz) \vee (\forall y \forall z \,\, x=yz+zy))$ and $\forall x \,\, x \cdot x = 0$? ...
0
votes
2answers
29 views

Zeroes of f(x) in a splitting field $E $ have the same multiplicity

Let $f(x)$ be an irreducible polynomial over a field $F$ and let $E$ be a splitting field of $f(x)$ over $F$. Then all the zeroes of $f(x)$ in $E$ have the same multiplicity. The proof of this ...
1
vote
0answers
50 views

Direct proof that $\mathbb{Z}/p\mathbb{Z}$ is not a flat $\mathbb{Z}/p^2\mathbb{Z}$-module.

I am trying to prove that $\mathbb{Z}/p\mathbb{Z}$ is not a flat $\mathbb{Z}/p\mathbb{Z}$-module. The reasoning I have is the following. We have an exact sequence $0 \to \mathbb{Z}/p\mathbb{Z} ...
0
votes
1answer
90 views

Will the Goldbach's conjecture be a law? [closed]

Till date, nobody (not even any supercomputer) has found any exception that violates Goldbach's conjecture. My math knowledge is up to K12 level. My questions are: Will the Goldbach's conjecture be a ...
1
vote
0answers
19 views

maximal chains of graded prime ideals

In Theorem 1.5.8 [Bruns,Herzog - Cohen-Macaulay-Rings] it is proved that for a noetherian graded ring $R$, a finitely generated graded $R$-module $M$ and any chain $\mathfrak{p}_0 \subsetneq ...
1
vote
1answer
84 views

Geometrically describe these Cosets and form a bijection with the Orbit-Stabilizer relation.

I am beginning to study abstract algebra/group theory and I have some seemingly simple practice questions here. I just want make sure I am understanding the concepts correctly. Here are the questions: ...
4
votes
2answers
103 views

Commutative and anticommutative ring that respects $x^2 = 0$, where $0$ is additive identity

Can anyone present an example where a ring consists of infinitely many different sets/elements in a universe (ring), is both commutative and anticommutative, and respects $x \cdot x=x^2 =0$ for ...
5
votes
1answer
66 views

group-like structure texts.

I was reading Dummit and Foote to be ready for my group theory text, but my teacher seems to be paying special attention to things with less structure than groups, for example monoids, semigroups, and ...
3
votes
0answers
46 views

Conjugacy classes of subgroup [closed]

Let $P$ be a $p$-Sylow subgroup in $G$. Prove that two elements in the center of $P$ are conjugate in $G$ if and only if they are conjugate in $N_G(P)$, the normalizers of $P$ in $G$.
0
votes
0answers
32 views

Taylor expansions in two variables

I need help in this proof Can I use Taylor expansion in Algebra? someone could give more detail of this Taylor expansion? Thanks in advance EDIT The main question is how the author get this "Y + ...
0
votes
0answers
19 views

$P$ is a simple point of $F$ $\Leftrightarrow O_P(F)$ is a DVR

I'm trying to find some sources with another proof of this theorem in Fulton's book: Does someone know other proofs of this theorem? maybe more algebraic? Thanks in advance
1
vote
2answers
57 views

A finite divisible group is trivial

I am having trouble seeing why a finite divisible group is necessarily trivial. Why does this have to be the case?
0
votes
1answer
35 views

Example of Tor-Rigid Module

Let $R$ be a commutative ring (with 1) and $M$ a finitely generated $R$-module. We say that $M$ is rigid if for every finitely generated $R$-module $N$ whenever Tor$_i^R(M,N)=0$ then Tor$_j^R(M,N)=0$ ...
1
vote
2answers
28 views

If $f:X\rightarrow Y$ is bijective, $A\subset X $ and $B \subset Y$, prove that $f(C_{X}A)=C_{Y}f(A)$

I want to prove the following proposition: "If $f:X\rightarrow Y$ is bijective, $A\subset X $ and $B \subset Y$, prove that $f(C_{X}A)=C_{Y}f(A)$". Obs>: the symbol $C_{X}$, $C_{Y}$ represents the ...
5
votes
0answers
129 views

Showing $\sqrt{2} \notin E.$

Let $E$ denote the least subring of $\mathbb{R}$ that is closed under the operation $r \mapsto e^r$. Then presumably, $\sqrt{2} \notin E.$ Question. How can we show this?
3
votes
2answers
154 views

Discriminant of $x^n-1$

Question is to find discriminant of polynomial $x^n-1$ I consider $f(x)=x^n-1=(x-a_1)(x-a_2)(x-a_3)\cdots(x-a_n)$ Now, ...
0
votes
0answers
30 views

Decomposition of UFD

Note that $R={\mathbb Z}$ is PID. So we have that if $I,\ J,\ K\subset R$ are ideals with $$ I = J\cap K$$ then $$(\ast)\ R/I = R/J\times R/K. $$ So we extend this. Let $R = {\mathbb Z}[x]$ which is ...
1
vote
2answers
112 views

Find all group homomorphisms from $\mathbb{Z}^n$ to $ \mathbb{Z}^n$

Can we find all group homomorphism from $\mathbb{Z}^n$ to $\mathbb{Z}^n$? For such a map surjective always imply isomorphism (like $\mathbb{Z}$)?