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.

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on primitive group actions with abelian stabilizers

I am trying to solve the following exercise from Dixon and Mortimer: Let $G$ be a finite primitive permutation group with abelian point stabilizers. Show that $G$ has a regular normal elementary ...
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1answer
71 views

Some residue field

Consider a prime ideal $\mathfrak{p}\in\mathrm{Spec} \ \mathbf{Z}[x]$; the residue field at $\mathfrak{p}$ is the fraction field of $\mathbf{Z}[x]/\mathfrak{p}$. Can we classify the residue fields? I ...
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23 views

Prove that $R[x]/(I,f)$ is isomorphic to $(R/I)[x]/(f')$, where $f'$ is $f$ in $R/I$.

Prove that $R[x]/(I,f)$ is isomorphic to $(R/I)[x]/(f')$, where $f'$ is $f$ in $R/I$. Attempt: I know how to prove $R[x]/(I) \cong (R/I)[x]$ using the first isomorphism theorem and the homomorphism ...
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0answers
25 views

Codimension of $\ker $ $\alpha $

Can someone explain why the codimension of $\ker $ $\alpha $ is $1$ in $ H $, with complement $ Fh_\alpha $? Is this because $ h_\alpha $ when $ \alpha $ is simple is part of the dual basis to ...
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+50

Meaning of notation $\operatorname{ord}_Q(g)$ in “Algebraic Curves” by Fulton

I didn't understand this notation in the chapter 7 page 93 of Fulton's algebraic curves book: What the author means by $\text{ord}_Q(g)$? Maybe he would like to say $\text{ord}_Q(G) := ...
4
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1answer
35 views

Let $K$ be a field extension of $F$ and let $a \in K$. Show that $[F(a):F(a^3)] \leq 3$

Let $K$ be a field extension of $F$ and let $a \in K$. Show that $[F(a):F(a^3)] \leq 3$. Find examples to illustrate that $[F(a):F(a^3)]$ can be $1,2$ or $3$. Attempt: $F \subset F(a^3) \subseteq ...
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0answers
45 views

A non-unital commutative ring with infinite elements such that each element $a$ satisfies $ab =0$ for infinitely many $b$'s

Beside the usual rules for non-unital commutative ring (that is, a ring without multiplicative identity) $R$, I want $R$ to satisfy the following: $ab = 0$ for each element $a$ and there are ...
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0answers
100 views

Multiplicity of a module in a particular case

We define multiplicity of a module M of dimension $d>0$ as $$mult(M) := lc (P_M) (d-1)!$$ where $P_M$ denotes the Hilbert polynomial of M. Equivalently, we have $mult(M) = Q_M(1)$, where $HP_M (z) ...
2
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1answer
22 views

A field extension of prime degree

Suppose that $E$ is an extension of $F$ of prime degree. Show that $~~\forall~ a \in E : ~ F(a)=F$ or $F(a)=E$ Attempt: Suppose that $E$ is an extension of a field $F$ of prime degree, $p$. ...
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48 views

What are subfields of $\mathbb{C}$?

I only took the first undergraduate abstract algebra course, so i don't know (at all) what Galois theory is about. I'm asking this question since i'm not sure of the definition of inner product space ...
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1answer
40 views

Another description of injective hull

Let $I$ be an injective module containing a module $M$, let $M_1$ be a submodule of $I$ maximal with respect to the property that $M_1∩M=0$, and let $M_2$ be a submodule of $I$ containing $M$ maximal ...
2
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1answer
30 views

Suppose that $f(x)$ and $g(x)$ are irreducible over $F$ and that $\deg f(x)$ and $\deg g(x)$ are relatively prime.

Suppose that $f(x)$ and $g(x)$ are irreducible over $F$ and that $\gcd(~\deg g(x),\deg f(x)~)=1$. If $a$ is a zero of $f(x)$ in some extension of $F$, show that $g(x)$ is irreducible over $F(a)$ ...
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47 views

Deduction of usual Cayley-Hamilton Theorem from “Determinant Trick”

Here is a statement of a standard theorem in commutative algebra (see page 60 of this book): Theorem. ("Determinant Trick") Suppose that $R$ is a commutative ring with $1$. Let $M$ be a finitely ...
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36 views

Dual spaces: Roots and Cartan subalgebra

Can someone show that the roots and the Cartan subalgebra are dual vector spaces? I don't see how simple roots acting on non-corresponding indices of a Cartan basis produce 0 and a simple root ...
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1answer
39 views

Jordan-Chevalley decomposition of $T$ acting on $k[T]/(\pi(T)^e)$

Given an algebraically closed field $K$, a f.d. vector space $V$ over $K$ and $A\in{\rm GL}(V)$, we can view the space $V$ as a $K[T]$-module, where $T$ acts by $A$. Using the fundamental theorem of ...
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1answer
22 views

Help understanding statement relating to structure of modules over PIDs

Lemma IV.6.11 of Hungerford's Algebra is Lemma 6.11. Let $R$ be a principal ideal domain. If $r \in R$ factors as $r = p_1^{n_1} \cdots p_k^{n_k}$ with $p_1,\ldots,p_k \in R$ distinct primes and ...
0
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1answer
46 views

If $f(N) = M $, $\ker f < N$, then the preimage of $M$ is contained in $N$? [closed]

This is from "Algebra" by Hungerford (page 44, proof from corollary 5.8) Let $f:G \rightarrow H$ be a homomorphism, $N$ be a normal subgroup of $G$, and $M$ be a normal subgroup of $H$. Show that if ...
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2answers
68 views

For a transitive permutation group $G,$ show that there is some $\sigma \in G$ such that $\sigma(a) \neq a$ for all $a \in A.$

Here is a problem that I have been working on. I was able to prove part A, but am having problems with part B. Thanks! Let $G$ be a permutation group acting on a finite set $A.$ If $g\in G,$ let ...
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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 $\Gamma(V)=k[X_1,\ldots,X_n]/I(V)$. ...
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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 ...
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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 ...
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1answer
60 views

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 ...
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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
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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
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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 ...
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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
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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 ...
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1answer
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$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 $$0\rightarrow {\rm Hom}_R(D,L)\rightarrow {\rm Hom}_R(D,M)\rightarrow {\rm Hom}_R(D,N)\rightarrow 0$$ ...
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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
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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
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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: ...
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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
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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 $F:B\to B$ ...
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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? ...
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205 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
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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 ...
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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 ...
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1answer
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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 ...
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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 ...
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2answers
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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?
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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: ...
3
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2answers
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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 ...
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28 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 ...
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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 ...
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0
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1answer
92 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 ...
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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
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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 $U_1 M = M U_2$?
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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
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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 ...