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)

1
vote
1answer
48 views

The intuition behind the coordinate ring $\Gamma(F)$

I'm studying Fulton's algebraic curves book. He gives the following definitions: We can define the coordinate ring of a nonempty variety $V\subset \mathbb A^n$ as $\Gamma(V)=k[X_1,\ldots,X_n]/I(V)$. ...
0
votes
2answers
64 views

Use Sylow's theorem to show that $G = HN_G(P)$

I am stumped on this question. Does anyone have some helpful hints or a solution to this question? Thanks! Let $G$ be a group of finite order. Let $H$ be a normal subgroup of $G.$ Let $P$ be a ...
2
votes
0answers
65 views

If $R$ is a domain and $M$ a finitely generated $R$-module, is it true that $\bigcap_{f\in M^{*}}\ker{f}=\operatorname{Tor}M$?

Let $R$ be a domain and $M$ a finitely generated $R$-module. Let $M^{*}=\hom_{R}(M,R)$. Let Tor$M$ be the torsion submodule of $M$. It it true that $$\displaystyle\bigcap_{f\in ...
1
vote
1answer
56 views
+100

Using resultants to check if multivariate polynomials have a common factor - is my proof correct?

Proposition: Let $f, g \in \mathbb R[x,y,z]$. Then the condition that $f, g$ have a common polynomial factor is an algebraic condition on their coefficients. By algebraic condition, I mean there is a ...
1
vote
0answers
80 views

Cardano, Descartes, Linear Equations and polynomials of degree greater than 3 [closed]

How would you describe the physical significance of algebraic equations of degree $> 3$, and generally of polynomials of degree $> 3$ ? One line of thought is that $x$ is length, $x^2$ is area ...
4
votes
2answers
103 views

Normalizer and centralizer of abelian subgroups of a group are equal [closed]

I have a question: Let $G$ be a finite group. If for each abelian subgroup $H$ of $G$ the centralizer and the normalizer of $H$ are equal, that is, $C_G(H)=N_G(H)$, prove that $G$ is abelian ...
3
votes
2answers
89 views

On Hilbert's Nullstellensatz Theorem

I was reading Ravi Vakil's notes on his website and he states the Hilbert Nullstellensatz (3.2.5.): If $k$ is any field, every maximal ideal of $k[x_1, ..., x_n]$ has residue field a finite extension ...
0
votes
0answers
43 views

Suggestions for choice of abstract algebra project [closed]

I want to know which of these topics have more materials and is interesting to write my project on... Group action as an extension of group multiplication or Wreath products of cyclic groups
1
vote
0answers
34 views

Is $\mathbb F_{p^e}^*$ cyclic for all $p$ and $e$? [closed]

I just came across a proof where it said that the (multiplicative) group of units of a certain finite field $\mathbb F_{p^e}^*$is cyclic. Is this true for all finite fields $\mathbb F_{p^e}$? Can ...
0
votes
1answer
34 views

$0\to L\to M\to N\to 0 $ is split if $0\to {\rm Hom}_R(D,L)\to {\rm Hom}_R(D,M)\to {\rm Hom}_R(D,N)\to 0$ is exact for any $D$.

Prove that $0\rightarrow L\rightarrow M\stackrel{\phi}\rightarrow N\rightarrow 0 $ is split if $$0\rightarrow {\rm Hom}_R(D,L)\rightarrow {\rm Hom}_R(D,M)\rightarrow {\rm Hom}_R(D,N)\rightarrow 0$$ ...
1
vote
1answer
37 views

Krull's theorem for a ring that does not have unit (multiplicative identitiy)

Is there some sorts of Krull's theorem (that every ring has maximal ideal) for rings that do not have multiplicative identity (unit)? So I know that non-unital rings do not satisfy Krull's theorem, ...
2
votes
1answer
42 views

Maximal homogeneous ideals of a graded $k$-algebra.

Let $k$ be an algebraically closed field, and let $A$ be a finitely generated commutative $k$-algebra. Given any maximal ideal $\mathfrak{m}\subset A$, we can form the quotient to obtain a map $A\to ...
2
votes
2answers
29 views

Nil radical of an ideal on a commutative ring

This is a problem of an exercise list: Let $J$ be an ideal of a commutative ring A. Show that $N(N(J))=N(J)$, where $N(J)=\{a \in A; a^n \in J$ for some $n \in \mathbb{N}\}$. What I did: ...
0
votes
2answers
59 views

Cohen-Macaulay and regularity

I know this is a simple question but to make sure....: $A$ is a commutative ring which is Cohen-Macaulay and for every maximal ideal $\mathfrak{m}$ in $A$ if $\dim A_{\mathfrak{m}}=\dim A$ then ...
2
votes
0answers
37 views

Kernels of induced maps in cohomology

Let $k$ be a field, and let $A$ and $B$ be two commutative $k$-algebras. Suppose $\varphi,\psi:A\to B$ are maps of $k$-algebras such that there are algebra automorphisms $G:A\to A$ and $F:B\to B$ ...
1
vote
2answers
64 views

$ o(ab)= o(ba)$, What about when $ab$ has infinite order?

I have been able to prove this when I say simply assume $o(ab)= n$, but I have been pondering the case of where $o(ab)$ is infinite? Is this something that needs to be considered? And if not, why? ...
5
votes
0answers
198 views
+100

Another Algebraic de Rham Cohomology question…

NOTE: scroll down to read my latest edit first if you're reading this for the first time :) My aim is to calculate the de Rham cohomology of the variety $U = \text{Spec} \ A$, where: $$A = ...
2
votes
1answer
57 views

Showing the Weyl algebra is simple.

Let $R$ be a ring with $1$, which contains $\mathbb{Q}$, and generated over $\mathbb{Q}$ by two elements $x,y$ such that $yx-xy=1$. Show that $R$ is simple. What i did? Certainly $x, y \in R$ as ...
2
votes
0answers
56 views

Question on a finite field extension of $\mathbb{Q}$

I have a polynomial $p(x) \in \mathbb{Q}[x]$ and is irreducible over $\mathbb{Q}$. Let it be of degree $n$ and $\alpha_1, ..., \alpha_n$ be its roots. I know that $$ \mathbb{Q}(\alpha_i) \cong ...
0
votes
1answer
21 views

Why this $F_*=F(X,Y,1)$

I'm studying Fulton's algebraic curves book. Someone could help me to prove this phrase highlighted: I didn't understand why the $F_*$ he defined is the same of the known $F_*=F(X,Y,1)$. Thanks ...
1
vote
2answers
73 views

Ring of rational-coefficient power series defining entire functions

I'm wondering if anyone has come across the following ring before. Let $R$ be the ring of complex power series $f=\sum_{n \ge 0} a_n t^n$ such that $a_n \in \mathbb{Q} \: \: \forall \: n$ The ...
-1
votes
2answers
66 views

Can $R \times R$ be isomorphic to $R$ as rings?

I know from this question that a ring $R \times R$ can be isomorphic to $R$, as $R$-modules. But can they ever be ismorphic as rings?
1
vote
2answers
32 views

How to find a basis of an image of a linear transformation?

I apologize for asking a question though there are pretty much questions on math.stackexchange with the same title, but the answers on them are still not clear for me. I have this linear operator: ...
2
votes
2answers
36 views

Kahler differentials and quotient rings.

I am dealing with some nice rings that are always isomorphic to some fairly nice quotient ring of a polynomial ring. A typical example is: $$ \mathbb{C}[X,XY,XY^2] \cong ...
0
votes
0answers
27 views
+100

Polynomial Ideals and Transverse modules

I have a given ideal and I want to find the "smallest" ideal so that when I add it to the original one, I get another certain ideal. Let $\mathcal{E}$ be the ring of smooth function germs ...
1
vote
3answers
83 views

The topology on $\mathbb A^2$ is not the product topology [duplicate]

I'm trying to prove the Zariski topology on $\mathbb A^2$ is not the product topology on $\mathbb A^1\times \mathbb A^1$. I'm looking for a counter-example based on the fact the closed subsets in ...
-2
votes
0answers
45 views
0
votes
1answer
91 views

Exercise about intersection of prime ideals

I need to write the ideal $I= (xz-y^2, yt- z^2) \subset R = \mathbb{K}[x,y,z,t]$ as intersection of prime ideals. Any idea? For the moment, I've noticed that I is radical, then it suffices to find ...
1
vote
1answer
22 views

Proof of ${\rm Hom}_R(R,R)=R$

If $R$ is commutative then $${\rm Hom}_R(R,R)=R$$ Proof : Let $A$ be an index set s.t. $$ R = \oplus_{a\in A} R_a $$ where $R_a$ is commutative Hence $$ {\rm Hom}_R(R,R) = \prod_{a\in A} {\rm ...
2
votes
2answers
51 views

Can units in $M_n(\mathbb{Z})$ be moved to the other side?

Let $M, U_1 \in M_n(\mathbb{Z})$ with $U_1$ a unit (i.e. $\lvert \det(U_1) \rvert=1$). Can I always find another unit $U_2\in M_n(\mathbb{Z})$ such that $U_1 M = M U_2$?
1
vote
0answers
75 views

Is group in which every $a$ satisfies $a^3=e$ abelian? [duplicate]

I know that any group in which every $a$ satisfies $a^2=e$ is abelian. How about if $a^3=e$ for every $a$?
3
votes
3answers
92 views

Finitely many embeddings of a finite extension in an algebraic closure

So I'm reading through Lang's Algebra, and he keeps saying something along the following lines: "Let $K$ be a finite extension of a field $k$ and let $\sigma_1,\ldots,\sigma_r$ be the distinct ...
1
vote
1answer
48 views

Inclusion of Fields whose order is a prime power

Blue was correct, I need to fix my understanding of this: Finite fields have cardinality of a prime order because they have a prime subfield that has finite characteristic. I do not completely ...
2
votes
3answers
109 views

What's $R[x]/(x^2)$ isomorphic to?

Can someone explain what the quotient ring $R[x]/(x^2)$ is isomorphic to? I know it's weird because it's reducible/has double root, but I'm not exactly sure what the implications of that, or how to ...
0
votes
1answer
68 views

defining the group law on elliptic curves in general

Let $k$ be an arbitrary field and $C \subset \mathbb{P}^2(k)$ an elliptic curve. In order to define the group law on $C$ we need to establish some geometric facts first, e.g. Any line intersects $C$ ...
1
vote
1answer
58 views

Quotient ring of $\mathbb{R}[x]$

I have learned in my class that quotient ring of $\mathbb{R}[x] / (x^2 + 1) \cong \mathbb{C}$. Just from curiosity, I was interested in knowing if $$ \mathbb{R}[x] / (x^2 + ax + b) \cong \mathbb{C} ...
0
votes
2answers
56 views

Existence of module homomorphism $\phi : M \rightarrow N$ such that $\Phi =\phi^2$

This is a homework problem and I have solved part (a), but I am not sure how to approach part (b). Should I approach it the way how the universal property of free modules are defined? Any hint(s) ...
2
votes
3answers
140 views

Showing Galois Group is Abelian

I'm having trouble showing that $\text{Gal}(\mathbb{Q}(\zeta_n)/\mathbb{Q})$ is Abelian. First I want to be able to show that $\mathbb{Q}(\zeta_n)/\mathbb{Q}$ is Galois, but I'm also not sure how to ...
0
votes
1answer
39 views

Remainder theorem for a real polynomial [closed]

A certain polynomial $p(x)\in\mathbb R[x]$, when divided by $x-a$, $x-b$, $x-c$ gives remainders $a$, $b$, $c$, respectively. How can I find the remainder when $p(x)$ is divided by $(x-a)(x-b)(x-c)$ ...
1
vote
0answers
35 views

Depth and kernel of multiplication map

If $A$ is a commutative $k$-algebra, $\mu: A\otimes_k A\rightarrow A$ is its multplication map and $I$ is the kernel of that map (viewed as an $A\otimes_k A$-module map) then what is the relationship ...
1
vote
0answers
50 views

Krull dimension of localization

If $R$ is a commutative ring and $m$ a maximal ideal therein, then what are the conditions for the Krull dimension of $R$ equaling to the Krull dimension of $R_m$?
0
votes
0answers
34 views

$2 {\mathbb Z}$ is flat

I want to prove that $2 {\mathbb Z}$ is flat. Note that this is not injective and not projective. Proof: For injective $ {\mathbb Z}$-module homomorphism $L \rightarrow_f M$, consider $$ 2{\mathbb ...
9
votes
3answers
117 views

Number of elements of order $2$ in $S_n$

How many elements of order $2$ are there in $S_n$? Using combinatorics I arrived at this: For $n$ even ($n=2k$) there are ${n\choose2}+{n\choose 2}{n-2\choose 2}\dfrac{1}{2!}+{n\choose 2} ...
0
votes
1answer
42 views

Irreducible ideals are prime in polynomial rings

Let $k$ be an algebraically closed field and $R$ the polynomial ring in $n$ variables over $k$. If $J$ is an irreducible ideal of $R$ then it is a prime ideal as well. To establish this statement ...
2
votes
1answer
39 views

Proof of Cauchy's Lemma in the case that G is abelian

I want to prove Cauchy's Lemma for abelian groups: If $G$ is abelian and there exists a prime such that $p$ divides the order of $G$, then there exists a $g \in G$ such that $p=\mathrm{ord}(g)$ I am ...
0
votes
2answers
42 views

Associative Lie algebra without Jacobi identity

1) Is there a name for associative Lie algebra that does not require Jacobi identity to hold? 2) Can such algebra exist, and if it does exist, can this algebra contain infinitely many elements? 3) ...
1
vote
1answer
25 views

Frobenius dihedral groups

How can we show by a direct group-theoretic proof that the dihedral group $D_{2n}$ is a Frobenius group iff $n$ is an odd number?
2
votes
1answer
37 views

Characteristically simple subgroup

Let $G$ be a finite group and $H$ be a minimal normal subgroup of $G$. How can we show that $H$ is a characteristically simple subgroup of $G$?
0
votes
0answers
17 views

Is there any semiring such that every element $z$ only has sum decomposition of $0+z$ and some another decomposition?

Let us say that there is a semiring $R$. By properties of semi-ring, every element $x \in R$ is equal to $0+x$. Is there any nontrivial semi-ring that every element $x \in R$ has only one other finite ...
1
vote
1answer
49 views

Multiplication by zero in an algebra over a field: $0x=0$ for every $x$?

If I have an algebra $A$ over a field $F$ and the zero element is $0\in A$. Is it true that $x0=0x=0$ for every $x\in A$? Thanks a lot!