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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Find a Four-element Abelian Subgroup of $S_5$ [duplicate]

Prof. Charles Pinter's "A Book of Abstract Algebra" provides this exercise: Ch 7 (Groups of Permutations) Part B #3 - Find a four-element abelian sub-group of $S_5$. Write its table. Please ...
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How to find inverse of generator of a finite field?

I need to find the inverse of generator of finite field $\mathbb{F}_{2^4}$ with irreducible polynomial , $f(x)=x^4+x+1$ i.e. if $g=0010$ is the generator of this field then how to find $g^{-1}$?
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Presentation of a group question

So I know that given a presentation of a group $G$, one can derive from the relations of the group presentation any element in the group $G$ right. However, I do have some confusion. If we take ...
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quaternions - understanding a formula

Quaternions are new for me. I am trying to understand the following formula: What are: $\large{q^x}$ ? I don't think it is a power. $\large{q^t}$ ? just a transposition of the quaternion $q$? ...
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1answer
38 views

Is there an element in $(\mathbb{Z}/7161\mathbb{Z})^*$ that has order 30? [duplicate]

I'm guessing that there is an element in $(\mathbb{Z}/7161\mathbb{Z})^*$ that has order 30, but can someone give me an example or a proof that there indeed is one? Either one is okay, whichever is ...
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1answer
22 views

Roots of $X^{l-1}+1$ in a quadratic extension $F_q$, $q=l^2$

Consider a finite field $F_q$ where $q=l^2$ ($l$ can be of the form $p^m$). Does $F_q$ has a root of $X^{l-1}+1$? As $X^l+X = X(X^{l-1}+1)$ we can show that $X^{l-1}+1$ splits if it has a root. This ...
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2answers
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What is an algorithm for describing the partition of this equivalence relation?

Let $\mathbb{R}$ be the set of real numbers,$ f : \mathbb{R} \to \mathbb{R}$ a map, and $E$ the equivalence relation on $ ℝ $ defined by $E = \{(x,y) \in \mathbb{R} \times \mathbb{R} \mid ...
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4answers
76 views

How to prove $x^p\equiv x\mod p$

I have already proved that $(x+y)^p\equiv x^p+y^p\mod p$. And if I take $y=1$, how do I prove the congruency $x^p\equiv x\mod p$? (I should use induction to $x$ but I dont know how to do that in ...
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Isomorphism type of the Galois group

$f=(x^2-2)(x^3-3)$. Let $K$ be the splitting field of $f$ over $\mathbb{Q}$. a) Determine the degree of extension of $K$ over $\mathbb{Q}$. b) Determine the isomorphism type of the Galois group of $K$ ...
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2answers
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Is group $G$ must abelian, when some condition is given by using exact sequence?

Suppose we are given the following exact sequence of groups where $A$ is an abelian normal subgroup of $G$: $$1 \rightarrow A \rightarrow G \rightarrow Q \rightarrow 1\tag{E}$$ If $G$ is Abelian, ...
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Basic application of the Nullstellensatz

Background: I have just started learning basic algebraic geometry. My solution to a simple problem involves an application of the Nullstellensatz and I want to know whether this is overkill (or ...
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0answers
47 views

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
56 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
34 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
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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 ...
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1answer
33 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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0answers
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
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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 ...
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1answer
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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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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 ...
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1answer
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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
37 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 ...
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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 ...
11
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5answers
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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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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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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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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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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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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
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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
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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
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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
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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: ...
0
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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
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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
119 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
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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
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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 ...
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1answer
35 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 ...
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1answer
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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 ...
4
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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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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
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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
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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?