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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Proof of Hensel's Lemma not clear

If you look at the following proof of Hensel's Lemma http://isites.harvard.edu/fs/docs/icb.topic1472247.files/Hensels%20lemma.pdf you will see that the author determines the conditions which these ...
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Intersection points of curves

In my lecture notes there is the following example for intersection points of curves: $$F(x, y, z)=xz^3-y^4 \\ G(x, y, z)=xz^2-y^3$$ in $\mathbb{P}^2(\mathbb{C})$, where ...
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1answer
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Two discrete lines always intersect at a point

In my lecture notes we have the following: $K$ field Extension of the affine space. Relation between points and lines: Two discrete points define an unique line and two discrete lines always ...
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2answers
48 views

Suppose $M$ is a Noetherian $R$-module and $\phi: M \rightarrow M$ is an $R$-module endomorphism of $M$.

So I did an exercise in my algebra textbook which was to show that $\ker(\phi^n) \cap \operatorname{im}(\phi^n) = 0$ and show that if $\phi$ is surjective, then $\phi$ is an isomorphism. I thought to ...
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2answers
75 views

The additive group of the reals is isomorphic to the additive group of the complex [duplicate]

Let $(\mathbb{R},+)$ be the additive group of the reals and $(\mathbb{C},+)$ be the additive group of the complex numbers. Prove that those groups are isomorphic. I think I got a solution using the ...
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1answer
73 views

$X$ is an infinite set. Prove that $S_X$ does not have proper subgroup of finite index.

Help Denote by $S_X$ the group of permutations of $X$, i.e. the group of bijections $f:X\to X$ with composition. Do we want to show $[S_X : H] = S_X?$
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1answer
14 views

field composition vs field extension generated by finite roots

is F(\alpha,\beta) the same field as F(\alpha)F(\beta)? one containment is obvious but I can't seem to prove the other way :S
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Checking if a quotient of a polynomial ring is a field

Show that $\mathbb{F}_2[X]/(x^3+x+1)$ is a field, while $\mathbb{F}_3[X]/(x^3+x+1)$ is not. If I'm right I just need to show that $x^3+x+1$ is irreducible in $\mathbb{F}_2$ and reducible in ...
2
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3answers
113 views

Field homomorphisms of $\mathbb{R}$

How to prove that $\mathrm{Hom}(\mathbb{R})=\mathrm{Aut}(\mathbb{R})$ ? (We treat it as field homomorphisms. ) I know that $\mathrm{Aut}(\mathbb{R})=\{\mathrm{id}\}$ and ...
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0answers
35 views

splitting fields $X^4-2,X^4+2X^2-2$

$L=\mathbb Q(\sqrt[4]{2},i)$ field extension of $\mathbb Q$. 1) Show $L$ is the splitting field of the polynomail $p:=X^4-2$ Now the roots of the given polynomial are $\pm\sqrt[4]{2},\pm ...
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39 views

Leads for penetrating the field of Algebraic number theory

I need to rapidly get up to speed on the following topics, for the purposes of an internship: Global and local fields. Localization. Number fields, function fields, etc. Ring of integers, field of ...
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1answer
24 views

Property of the intersection multiplicity: $I(P, y \cap x)=1$

How can we show the following property of the intersection multiplicity? $$I(P, y \cap x)=1 , \text{ where the point } P \text{ is at } (x, y)$$ Edit: My try: $$I(P, f \cap g ) \geq m_P(f) ...
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1answer
30 views

Finding an isomorphism of groups

Let $R$ be a commutative ring, $R_U$ be the group of units of R. Show that (i) $\mathbb{C}_U \cong \left(\left(\mathbb{R},+\right)/\mathbb{Z}\right)\times\left(\mathbb{R},+\right)$ ...
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2answers
18 views

Showing the inverse map of a ring homomorphism of a prime ideal is again a prime ideal

Let $\phi : A \rightarrow B$ be a ring homomorphism and $I$ be a prime ideal of $B$. (i) Show that $\phi^{-1}(I)$ is a prime ideal of $A$, and (ii) find an example of $A$, $B$ and $I$ so that ...
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2answers
36 views

Showing a factor ring is a field if it is an integral domain

Let $K$ be a field, $0 \neq I \subsetneq K[x]$ be an ideal. Show that $K[x] \ / \ I$ integral domain $\Rightarrow$ $K[x] \ / \ I$ field applies. My idea was to show that $K[x] \ / \ I$ is a ...
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1answer
17 views

Subfield generated by a multiplicative subgroup of the field

Let $F$ be a field with prime $p$ characteristic and let $X$ be a periodic subgroup of $(F,\, \cdot)$. Let now $K$ be the subfield of $F$ generated by the elements of $X$. Is it possible to describe ...
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15 views

Properties of intersection multiplicity

I am reading the properties of the intersection multiplicity and in my lecture notes there is the following: We have $f(x, y) \in \mathbb{C}[x, y], g(x, y) \in \mathbb{C}[x, y]$ and $P=(a, b) \in ...
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0answers
17 views

Direct sum of ideals over Dedekind domain [duplicate]

I'm trying to show that Let $\frak{a},\frak{b}$ be two ideals of a Dedekind domain $\cal{O}$. Show that there is an isomorphism \begin{equation*} ...
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1answer
20 views

Characterizing the Prüfer $p$-group

I've been trying to solve these questions for the past few hours with no luck: If $G$ is an infinite abelian group all of whose proper subgroups are finite, then $G$ is a Prüfer $p$-group for some ...
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3answers
67 views

Writing elements of field extension in terms of the basis determined by a root of a polynomial

Let $\alpha \in \mathbb{C}$ be a root of the irreducible polynomial $$f(X) = X^3 + X + 3$$ Write the elements of $\mathbb {Q}(\alpha)$ in terms of the basis $\{1, \alpha, \alpha^2\}$. The first ...
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1answer
54 views

$G$, $|G|=n$ is nilpotent $\iff$ $\forall m|n$, $G$ has a normal subgroup of order $m$.

Prove $G$, $|G|=n$ is nilpotent $\iff$ $\forall m|n$, $G$ has a normal subgroup of order $m$. I got stuck in the second direction. One direction: $|G|=n=p_1^{s_1}\cdot ...\cdot p_k^{s_k}$ Where $p_i$ ...
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2answers
22 views

How is the resultant defined?

In my lecture notes we have the following: We have that $f(x, y), g(x, y) \in \mathbb{C}[x, y]$ $$f(x,y)=a_0(y)+a_1(y)x+ \dots +a_n(y)x^n \\ g(x, y)=b_0(y)+b_1(y)x+ \dots +b_m(y)x^m$$ The ...
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1answer
32 views

Do involutions in the symmetric group form a basis of a Jordan algebra?

Let $M_d$ is a vector space spanned by the set of all the involutions in the symmetric group $S_d$ (you can treat $M_d$ as a space of function on the set of involutions endowed with standard addition ...
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1answer
38 views

For a commutative ring $R$, why does $1-ab$ being a non-unit leads to $1-ab \in M$ for some maximal ideal $M$?

Suppose there is a commutative ring $R$, without any restriction. Now suppose $a,b \in R$. If $1-ab$ is a non-unit, why is there at least one maximal ideal $M$ that $1-ab \in M$?
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1answer
47 views

Is $(X^3 - 18X + 12, 5) \in \mathbb{Z}[X]$ a prime ideal?

I'm trying to determine wheter $A = (X^3 - 18X + 12, 5)$ and $B = (X^3 - 18X + 12, X-1)$ is a prime ideal in $\mathbb{Z}[X]$ and $\mathbb{Q}[X]$. I know that $A = \mathbb{Q}[X]$ since I can make ...
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1answer
37 views

For a non-unit element $x$ in a unital ring, does non-zero $a$ or $b$ ALWAYS exist s.t. $ax=xb=0$? [on hold]

The question is as given in the title: For a non-unit element $x$ in a unital ring, does non-zero $a$ or $b$ always exist s.t. $ax=xb=0$?
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1answer
42 views

Question related to commutative ring being Noetherian

Let $A$ be a commutative ring with $1$, and $A = (f_1, \ldots, f_n)$. I want to prove the following: If $A$ is a Noetherian ring, then so is $A_{f_i}$ (which is the ring $A$ localized at $f_i \in ...
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PBW for Lie Superalgebras

The PBW Theorem In the literature there are many sources discussing the PBW-basis for Lie superalgebras, see for example M. Scheunert - The Theory of Lie Superalgebras theorem 1 and corollary in ...
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46 views

What is the difference between $[H, g]$ and $[h, g]$?

I am working on this problem, where $[H, g]$ is the commutator group: Let $H$ be a subgroup of $G$, show that $[H, g] = [H, \langle g \rangle]$. Before solving it, I need to understand the ...
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4answers
63 views

An elementary fact about tensor product of modules

Let $A$ be an $R$-right module, $N$ be a submodule of $R$-left module $M$, $\pi:M\rightarrow M/N $ is the natural epimorphism. What is $\ker\left(\pi\otimes1_{A}\right) $? I appreciate your help.
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Is integrally closed domain finitely generated?

Does integrally closed domains have finite number of generators that generate the whole ring?
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2answers
56 views

Show that $\mathbb{A}^n$ on the Zariski Topology is not Hausdorff, but it is $T_1$

There was an exercise I could not do. So the property is $T_1$ if for every pair of distinct points, $P, Q \in X$, there is an open subset $U$ containing $P$ but not $Q$ and another open subset $V$ ...
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1answer
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let E be a field, f is a ring homomorphism:E to E, so f is a isomorphism. [on hold]

let $E$ be a field, f is a nontrivial ring homomorphism:$E \rightarrow E$. Prove that $f$ is a isomorphism.
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Irreducible components of the curve-Algebraic set

In my lecture notes I have the following: $f, g \in \mathbb{C}[x,y]$ $V(f)=V(g)$ if $f=p_1^{a_1} \cdot p_2^{a_2} \cdot \dots \cdot p_s^{a_s} , g=p_1^{b_1} \cdot p_2^{b_2} \cdot \dots \cdot ...
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1answer
58 views

Proof $27X^3 - 13X^2 + 180 \in \mathbb{Q}[x]$ is irreducible without Gauss's lemma

I'm asked to show that $27X^3 - 13X^2 + 180 \in \mathbb{Q}[x]$ is irreducible in $\mathbb{F}_{13}[X], \mathbb{Z}[X]$ and $\mathbb{Q}[X]$. I've managed to proof the first two. In $\mathbb{F}_{13}[X]$ I ...
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2answers
53 views

A general question about Isomorphism Abstract Algebra

I want to ask a general question about $p^2$-groups. How can I know if a group is isomorphic to $\mathbb{Z}_{p^2}$ or to $\mathbb{Z}_p \times \mathbb{Z}_p$ ? Thanks
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1answer
46 views

homomorphisms of field extension

Let $\bar{\mathbb Q}$ be an algebraic closure of $\mathbb Q$. Determine all homomorphisms from $\mathbb Q(\sqrt[4]{2},i)\rightarrow\bar{\mathbb Q}$ and their images! Now the minimal polynomials for ...
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1answer
27 views

Finitely Generated Sets

Let $F$ be a field and suppose $I$ is an ideal of $F[x_i, \ldots , x_n]$ generated by a (possibly infinite) set $S$ of polynomials. Prove that a finite subset of polynomials in $S$ is sufficient to ...
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1answer
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Nullstellensatz: If $V(f)=V(g)$ we have that $Rad \langle f \rangle =Rad \langle g \rangle$

In my lecture notes I have the following: From the Nullstellensatz (NSS for short) we have the following: $$\text{ If } V(f)=V(g) \Rightarrow V(Rad(\langle f \rangle ))=V(Rad \langle g \rangle ) ...
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1answer
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Is ring R itself a finitely generated module over $R$?

It seems trivial that ring $R$ itself is a $R$-module. But then can we say R is finitely generated by multiplicative identity? That seems so trivial..
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Definition of minimal presentation of a group

I'm working on a problem on the braid monodromy of complex lines arrangements in $\mathbb{C}^{2}.$ I have the following question. It's just a simple definition. However, I didn't find anywhere. Let ...
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1answer
49 views

Sum of ideals-Intersection of algebraic sets

In my lecture notes I have the following: $$ \begin{array}{ccl} \text{Sum of ideals} & & \text{Intersection of algebraic sets} \\[4pt] I+J & \longrightarrow & V(I+J)=V(I)\cap V(J) \\ ...
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2answers
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Do minimum polynomials always have a nonzero discriminant?

Let $f(x)$ be a minimum polynomial with integer coefficients. Does $f(x)$ Always have its discriminant equal to nonzero ? If so , why can't $f(x)$ have a repeated root ? For all clarity Im talking ...
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33 views

Orbits of left-multiplication from $\mathrm{PSL}_2(\mathbb Z)$ on $\mathbb Z^{2\times 2}$

I am trying to learn moduli space of elliptic curves from different resources. But only a spacial class of elliptic curves where the lattice in the plane has "integer vectors" as generators. To study ...
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108 views

If G is a group of even order, prove it has an element $a \neq e$ satisfying $a^2 = e$.

If G is a group of even order, prove it has an element $a \neq e$ satisfying $a^2 = e$. My proof: Let $|G| = 2n$. Since G is finite, there exists, $a \in G$ such that $a^p = e$ and by Lagrange's ...
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Find $n$ s.t. $Aut(\mathbb Z_2 \oplus \mathbb Z_4 \oplus \mathbb Z_4 \oplus \mathbb Z_6) \cong U(n).$

can we determine the automorphism group of a $U$-group i.e $Aut(U(n))$ ? I need to find $n$ s.t $Aut(\mathbb Z_2 \oplus \mathbb Z_4 \oplus \mathbb Z_4 \oplus \mathbb Z_6) \cong U(n).$ ? I started by ...
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Infinite group G is polycyclic then subgroup of fitting is nilpotent.

Let G be a infinite polycyclic group i.e soluble and satisfy max. Show that the subgroup of fitting of G is nilpotent. he subgroup of fitting of G is subgroup of G generated by normal nilpotent ...
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72 views

Classification of indecomposable modules over a given ring

Let $K$ be a field, how can I classify all the indecomposable modules up to isomorphism over the ring $R = K [x] / (x^n ) $?
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1answer
49 views

Is the set of continuously differentiable functions on [0,1] an integral domain?

Is the set of continuously differentiable functions on $[0,1]$ an integral domain? I considered the functions $f, g$ defined by $f(x)= \left\{ \begin{array}{cc} 0, & 0 \leq x \leq \frac{1}{2};\\ ...
3
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1answer
32 views

Integral closed domain and localization of $\mathbb{Z}$ respect to prime ideal

We know that $\mathbb{Z}$ is integrally closed domain. This means that with respect to its prime ideal $p$, localization $\mathbb{Z}_p$ is also integrally closed in its field of fractions. Suppose ...