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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Is every reduced $k$-algebra all of whose residue fields are $k$ finitely generated?

Let $k$ be a field (of characteristic zero if you want). Let $A$ be a reduced $k$-algebra with the property that for every prime ideal $\mathfrak{p}$ of $A$ the natural homomorphism $k \to A/ ...
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1answer
24 views

Polynomial irreducibiliy with substitution (need evaluation of logic)

One thing I have seen several times when trying to show that a polynomial $p(x)$ is irreducible over a field $F$ is that instead of showing that $p(x)$ is irreducible, I am supposed to show that $p(ax ...
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2answers
58 views

Proving $0x=0$ in a ring

I am trying to prove the above trivial statement. I am aware of the standard approach of letting $0 = 0 + 0$ and cancelling, but I would like the below statement to be verified/corrected: $1\cdot ...
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1answer
35 views

Zorn's lemma converse? (Context: Maximal proper subgroups)

So, in my qual prep class a pretty simple question popped up: "Prove that for any nontrivial finite group there exists a maximal proper subgroup." So of course, my natural inclination was to ...
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1answer
26 views

Polynomial Rings

I know that an Euclidean ring is a principal ring and this last is a factorial ring. I also know that if $B$ is a field, then $B[x]$ is Euclidean. While, for instance $\mathbb{Z}[x]$ is not a ...
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2answers
40 views

Finite fields, characteristics and the Fundamental Homomorphism Theorem

I am trying to make sense of this proposition. I am fine with part (a), for part (b) however, can you explain what the computation proves? Can you not verify a homomorphism by checking the 3 ...
2
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1answer
34 views

About the ways prove that a ring is a UFD.

I'm doing an exercise where I have a commutative ring with unity $R$. We had to find that the nonunits formed an ideal (maximal). After that, we found the irreducible elements, and then we saw that ...
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24 views

Purely transcendental proper extension not algebraically closed? [on hold]

I'm having trouble proving this Dummit and Foote exercise: Prove that a purely transcendental proper extension of a field is never algebraically closed.
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1answer
17 views

Is normal extension algebraic?

Let $E/F$ be a field extension. Then $E/F$ is called normal iff there exists $\mathscr{A}\subset F[X]\setminus\{0\}$ such that $\forall f\in\mathscr{A}$, $f$ splits over $E$ and ...
2
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1answer
33 views

If some non-primitive polynomial in $\mathbb Z[x]$ is irreducible over $\mathbb Q$, does this imply it is irreducible over $\mathbb Z$?

I know from a lemma in Herstein that if a primitive polynomial in $R[x]$ is irreducible in its field of quotients $F[x]$, then it is irreducible in $R[x]$. But, if some non-primitive polynomial in ...
3
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1answer
35 views

Show that the ring $R$ of entire functions does not form a Unique Factorization Domain

Show that the ring $R$ of entire functions does not form a Unique Factorization Domain (U.F.D) My try: I will first check whether $R$ forms an Integral Domain then check whether it is Factorization ...
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4answers
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If $G$ is cyclic then $G/H$ is cyclic?

If $G$ is cyclic, then $G/H$ is cyclic? The proof I got goes like this: $G$ is cyclic, so $G=<g>$ for some $g\in G$. So any coset in $G/H$ would be of the form $Hg'=Hg^n$ for some $n$. So ...
2
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1answer
33 views

Division algorithm for polynomials

When we do the division algorithm for polynomials, how do we figure out $ca^{-1}$; i.e., for the problem where $f(x)= 3x^2+2$ and $g(x)= 4x^4 + 2x^3 + 6x^2 + 4x + 2$ in $\mathbb{Z}_7[x]$. Here, $a= ...
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1answer
38 views

If $X \subseteq G$, $\langle X^{G}\rangle$ is normal in $G$

Let $X^{G} = \{gxg^{-1}: g \in G, x \in X\}$, and define $$\langle X^{G}\rangle = \bigcap_{H \in A} H,$$ where $A=\{H \leq G: X^{G} \subseteq H\}$. We wish to show that $\langle X^{G}\rangle$ is ...
11
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3answers
727 views

Are there more groups than rings?

It seems pretty clear to me that both of these are at least uncountable (which I think I could prove with some work). It also seems that you should be able to make some diagonal argument about the ...
12
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5answers
175 views

Is 'Algebraic Number Theory' the study of the theory of algebraic numbers, or is it the study of the theory of numbers from an algebraic viewpoint?

Is Algebraic Number Theory the study of the theory of algebraic numbers? Or is it the study of the theory of numbers from an algebraic viewpoint? Or is it both? I know I can just find a wiki article ...
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0answers
25 views

Connecting homomorphism of exact sequence of Ext groups

Let $0\to M\to L\to N\to0$ be an exact sequence of modules over a ring $A$. Having an $A$-module $K$ we obtain the exact sequence of Ext groups $$0\to Hom_A(N,K)\to Hom_A(L,K)\to ...
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3answers
45 views

Identity element of a group

So I already proved Closure and Associativity, now I'm trying to find the identity element of this operation defined as: $$ a * b = a + b - ab $$ But my identity element gets cancelled... (The set ...
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0answers
30 views

Deriving that center is subset of centralizer(using group actions only)

So if we consider our group $N_G(A)$ and let the group act on the set A $\in$ P(G) via conjugation and consider the kernel we will get precisely the kernel being the centralizer, and since the kernel ...
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0answers
31 views

Determine the galois group of a quartic

I'm reading Hungerford's algebra chapter about galois theory. There is the following theorem in p.273 (with some minor changes) about determining the Galois group of a quartic: Let $K$ be a field ...
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3answers
250 views

Geometrically, why do line bundles have inverses with respect to the tensor product?

Geometrically, why do line bundles have inverses with respect to the tensor product? Here my thoughts on the problem so far, please excuse their scatteredness. I know algebraically, it is just ...
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1answer
46 views

What does the ring $R=C[x]/I$ look like?

Maybe it's a stupid question but what does the ring $R=C[x]/I$ look like? $I$ is the ideal in $C[x]$. Everything helping! Thanks :)
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1answer
36 views

Does an orthogonal decomposition of a vector space exist?

Let V be a complex vector space equipped with an hermitian form (not necessarily positive definite), W a finite dimensional subspace of V such that it has zero radical (intersection between W and its ...
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0answers
55 views

How much algebra is necessary to understand Rudin's “Real and Complex Analysis”?

I've been reading up on the finite element method, and the text many people recommend is The Mathematical Theory of Finite Element Methods by Brenner and Scott. As part of the background, the authors ...
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how to find solution(all points ) of $y^2+xy=x^3+x^2+1$? [on hold]

actually ,this is elliptic curve(koblitz curve) and I want to know all the points on this curve. so would you please tell me how to find points on this curve defined over finite field(F 2^4)??
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0answers
48 views

Meromorphic functions on $Y^2 = X^3 + 1$, genus.

Let $k$ be a field of characteristic $\neq 2$, and consider the quadratic extension $F$ of $k(X)$ generated by $\sqrt{X^3 + 1}$. What is/how do I find the genus of $F$? The progress I have so far: ...
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1answer
49 views

Proof of No Unique Factorisation in $\mathbb Z[\sqrt{d}]$ for $d \leq-3$

How would I prove there is no unique factorisation in $\mathbb Z[\sqrt{d}]$ for $d \leq-3$, where $d$ is a square-free integer? I think it's something to do with the only invertible elements ...
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2answers
32 views

Prove that $p(x)$ is irreducible in $F[x]$

Let $F$ be a field and let $K$ be an extension of $F$. Let $\alpha$ be algebraic over $F$. Let $p(x)$ be the polynomial of minimal degree having $\alpha$ as a root. Prove that $p(x)$ is irreducible in ...
10
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1answer
123 views

DVR, power series expansion.

Let $A$ be a discrete valuation ring with quotient field $K$, maximal ideal $\mathfrak{m}$, uniformizing parameter $t$. Let $k = A/\mathfrak{m}$, so $k$ is a field. How do I show that there is a ...
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2answers
48 views

Extension of prime ideals in Dedekind domains

In various textbooks and lecture notes on algebraic number theory, I have found the following claim without proof: Let $R$ be a Dedekind domain with field of fractions $F$ and let $S$ be its integral ...
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1answer
46 views

Only DVR's with quotient field $\mathbb{Q}$?

Let $p \in \mathbb{Z}$ be a prime number. I know how to show that $$\{r \in \mathbb{Q}: r = {a\over{b}},\text{ }a,b \in \mathbb{Z},\text{ }p\text{ doesn't divide }b\}$$ is a DVR with quotient field ...
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2answers
40 views

Prove equivalence between $X$ Hausdorff and $X$ finite with discrete topology

We have a Noetherian topological space $X$. Show that the following are equivalent: $X$ is a Hausdorff space $X$ is finite and has discrete topology So far I've only got this: If $X$ has discrete ...
0
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1answer
38 views

Flat Module finitely generated when over the residue field finite dimensional? [on hold]

Let $(A, \mathfrak{m})$ be a local ring with residue field $\kappa=A/ \mathfrak{m}$. Let $M$ be a flat $A$-module. Assume that $M \otimes_A \kappa$ is a finite dimensional $\kappa$-vector space. Is it ...
5
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1answer
91 views

Complement of open set is finite in Zariski topology

This problem has two parts: a) Let $M$ be a finitely generated module over a Noetherian ring $A$. Prove that $S=\{ P \in\operatorname{Spec}(A) : M_P \mbox{ is a free }A_P\mbox{-module} \}$ is an ...
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1answer
26 views

Local ring coincides with DVR.

Assume $A$ is a discrete valuation ring with quotient field $K$ and maximal ideal $\mathfrak{m}$. If $S$ is a local ring containing $A$ and contained in $K$ with maximal ideal containing ...
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23 views

Possible values of the GCD of two polynomials

Let $p(x)$ be a polynomial in $\mathbb Q[X]$. Find the possible values of $d=gcd(p(x),p(x)+x-1)$. I have: $gcd(p(x),p(x)+x-1)=gcd(p(x),x-1)$ Is the answer to the question: $d(x) \in ...
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1answer
27 views

Prove that every Principal ideal Domain is a Unique Factorization Domain

I know that to prove something is a a Unique factorization Domain i need to show that the factorization is unique. So i start like that.// Proof: Let $P$ be a principal ideal domain, and let $$r \in ...
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2answers
35 views

If $φ:K → Aut(H)$ is the trivial homomorphism—then the semidirect product is just the ordinary direct product of groups.

I'm working on the question Show that if $φ:K → Aut(H)$ is the trivial homomorphism—i.e., $φ(k) = 1_H$ for every $k ∈ K$— then the semidirect product is just the ordinary direct product of groups. I ...
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2answers
26 views

Quotient rings, polynomials are reducibility

I am trying to follow this solution. I am struggling to understand why 'If g is a member of R, then g divides the content of f'. Why is this true?
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1answer
39 views

computing in a loop

Consider a loop, i.e. a set $Q$ with operation $\cdot$ such that we have cancellation law and an identity element. For given $x,y\in Q$ consider the equation $$x=((xy)^{-1}_R\cdot z)^{-1}_L$$ where ...
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0answers
13 views

What are all the $k$ dimensional unimodular subspaces of $\mathbb{Z}^n$?

I am trying to prove the following assertion - The set of subgroups of $(S^1)^n$ which are isomorphic by an element of $Aut((S^1)^n)$ to the standard copy $(S^1)^k$ is naturally parametrized by the ...
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1answer
29 views

Prove the universal property of free abelian groups

I want to prove the following: Given a set $S$ show that the free abelian group (free group quotient the commutator) generated by $S$ is the smallest abelian group that contains $S$. The thing is ...
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Principal ideal domains and polynomial rings [duplicate]

As is well known if $\mathbb F$ is a field, then the polynomial ring $\mathbb F[x]$ is a principal ideal domain. Now, let $R$ be a commutative ring with identity and assume that the polynomial ring ...
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3answers
575 views

Short exact sequence - Why doesn't this argument work?

What is wrong with this "proof"? If the sequence of $\Bbb{Z}$-modules $$0\to M \to N \to \Bbb{Z}/2 \to 0$$ is exact, then $N\cong M \oplus \Bbb{Z}/2$. Call the first map $f$, the second $g$. By ...
6
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1answer
45 views

Gauss's lemma: More than a stepping stone on the way to proving $R[x]$ is a UFD when $R$ is?

I'm reviewing my abstract algebra a bit. Currently looking at UFDs. In this context, Gauss's lemma (or part of it, at least) says that the product of two primitive polynomials over a UFD is primitive. ...
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0answers
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Is my understanding of this corollary correct?

The following is a theorem/corollary pair in an introductory abstract algebra course. Theorem: $f(x)\equiv g(x) $ mod $p(x)$ if and only if $[f(x)]=[g(x)]$, where $[h(x)]=h(x)$ mod $p(x)$. ...
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3answers
32 views

Possible textbook redundancy concerning invertible mappings

In my textbook (Modern Algebra by John Durbin, 6th Ed), there is the following theorem: Let $S$ denote any nonempty set. (a) Composition is an associative operation on $M(S)$, with identity ...
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2answers
63 views

polynomials root finding [closed]

Is every root of a polynomial of positive integer degree n, and with a rational coefficients is considered algebraic number? and how one can find some roots to this polynomial ...
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1answer
26 views

Homomorphism on U(36)

Question Suppose that $f$ is homomorphism of $U(36)$, $\ker(f) = \{1,13,25\}$, and $f(5) =17$. Determine all the elements that map to 17. What I've tried so far So I've determined that $U(36) = ...
3
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1answer
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Modular Forms: Find a set of representatives for the cusps of $\Gamma_0(4)$

We write $SL_2(\mathbb{Z})$ as $\coprod\alpha_i\Gamma_0(4)$, where $\coprod$ means the disjoint union, and the $\alpha_i$ are the coset representatives of $SL_2(\mathbb{Z})\diagup\Gamma_0(4)$. We ...