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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Ideal of a product ring?

I am trying to prove whether or not the ring generated by $\langle (2,2)\rangle$ is a prime idea of $\mathbb Z_4\times \mathbb Z_4$? My issue is I'm not sure how to do the coordinate multiplication: ...
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8 views

$R_{a} = R[x]/(x)$ isomorphic to $R_{b} = R[x]/(x-1)$

I am looking at the following two rings: $R_{a} = R[x]/(x)$ and $R_{b} = R[x]/(x-1)$. I was told that these two rings were isomorphic, but I don't see why. Is this due to the minimal polynomials? ...
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1answer
15 views

Counting permutations in $S_n$ with $1,2,..,k$ all in same cycle

The number of permutations in $S_n$ for which the first $k$ items $1,2,...,k$ are all in the same cycle can be shown (by a somewhat tedious argument) to be $n!/k.$ I'm looking for less computational ...
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1answer
27 views

Prove $Q[x]/(x^2+4)$ is isomorphic to $Q[x]/(x^2+1)$

I've been asked to prove Q[x]/(x^2+4) is isomorphic to Q[x]/(x^2+1); I've looked at lots of similar solutions, but haven't been able to understand this. I know each ring is the quotient ring for their ...
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5 views

Endowing module with algebra

Given a module $M$ over a ring $R$, is it possible to endow $M$ with an operation $M^{2} \to M$ that turns $M$ to an algebra that is not the trivial $m n \equiv \mathbf{0}$ identically? So far I have ...
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1answer
26 views

Tensor Product of $\mathbb{Q}[\sqrt{2}]$.

How can one show that $\mathbb{Q}[\sqrt{2}] \otimes_{\mathbb{Q}[\sqrt{2}]} \mathbb{Q}[\sqrt{2}] \simeq \mathbb{Q}[\sqrt{2}]$ (which is a $2$ dimension vector space over $\mathbb{Q}$) and ...
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11 views

Characteristic of a product ring?

Let A and B be commutative rings with unity where char(A)=n and Char(B)=m s.t. n,m ∈ ℤ (and n,m ≠ 0). Prove or give counter example: if k ∈ ℤ+ and n,m both divide k, then Char (A x B)| k. Here was ...
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30 views

Proof of the fixed point theorem in group action.

$\textbf{Fixed point theorem:}$ Let $G$ be a $p$-group and let $S$ be a finite set on which $G$ operates. If the order of $S$ is not divisible by $p$, there is a fixed point for the operation of $G$ ...
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1answer
22 views

abstract algebra: finite fields and galois group

I'm trying to find the order and all generators of the Galois group of $GF(256)$ over $GF(16)$? How to approach this problem?
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12 views

nondegenerate bilinear form $\mathrm{dim}{S}+\mathrm{dim}{S^{\perp}}=n$

I was told that in a linear space $V$ with nondegenerate bilinear form$\langle\cdot,\cdot\rangle$ , and $S$ is a subspace of $V$. we have $$ \mathrm{dim}{(S)}+\mathrm{dim}{(S^{\perp})}=n $$ where ...
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There are at most finitely many square-free integers $d\not\equiv 1\pmod{4}$ such that $\mathbb{Q}(\sqrt{d})$ is an euclidean field

My book's exercise is about proving that there are at most finitely many square-free integers $d\not\equiv 1\pmod{4}$ such that $\mathbb{Q}(\sqrt{d})$ is an euclidean field (with respect to the norm). ...
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54 views

Galois extension of degree $2^n$

Let $k$ be a field and $L=k(\sqrt{a_1},\sqrt{a_2},\cdots,\sqrt{a_n})$ with $a_i\in k$. Suppose $[L:k]=2^n$. Let $a=\sqrt{a_1}+\sqrt{a_2}+\cdots\sqrt{a_n}$. Show that $L=k(a)$.
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115 views
+100

Number of generators of prime ideals in $K[x_1,x_2,…,x_n]$

Is there any bound for the number of generators of prime ideals in $K[x_1,x_2,...,x_n]$? (For example in $K[x,y]$.) We know that maximal ideals of $K[x_1,x_2,...,x_n]$ have $n$ generators.
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21 views

Basis of irreducibles of non algebraic lattices

Let $L$ be a complete lattice. An element of lattice $x$ is called $\wedge$-irreducible if $x=y\wedge z$ implies $x=y$ or $x=z$. Similarly, it is completely $\wedge$-irreducible if $x=\bigwedge x_i$ ...
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29 views

Intersection of two principal ideals is an ideal and lowest common multiple (if it is a PI)

I think the first part of the proof would go like this: any element in $(a) \cap (b)$ can be written as $ar_1 = br_2$, so multiplying by an element $r \in R$ yields $ar_1r\in aR$ or $br_2r \in bR$, so ...
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28 views

Generators of two groups with prime order $p$ already induce all the generators of the product group $G \times H$

Let $G = \langle g \rangle, H = \langle h \rangle$ be two cyclic groups (with $g \in G, h \in H$), both of them of order $p \in \mathbb{N}$, where $p$ is a prime number. I now want to show that ...
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If $K=F(K^p)$ is a finite extension and $\{a_1,\ldots,a_n\} \subset K$ linearly independent then so is $\{{a_1}^p,\ldots,{a_n}^p \}$

Suppose that $F$ is a field of characteristic $p$. Let $K/F$ be a finite extension and $K=F(K^p)$, where $K^p:= \{x^p\mid x\in K\}$. Suppose $\{a_1,\ldots,a_n\} \subset K$ is linearly independent ...
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22 views

If $d = (a, b)$ and $a=da_1$, $b=db_1$, show that $(a_1, b_1)=1$. [duplicate]

First off, the problem states that $d$ is the GCD of $a$ and $b$ and $a_1$ and $b_1$ are integers. Now I tried putting them in a linear combination such that it would look like $da_1x+db_1y=d$. Well ...
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1answer
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Group of translations is a normal subgroup of group $E(2)$ of isometries of $\mathbb{R}^2$?

I'm trying to see that: The group of translations $T=\{t(x)=x+a : a \in \mathbb{R}^2 \}$ is a normal subgroup of group $E(2)$ of isometries of $\mathbb{R}^2$. I know the definitions of a normal ...
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1answer
16 views

What's group $E(2)$ of isometries of $\mathbb{R}^2$?

I'm trying to prove normal subgroups of the group $E(2)$, but I haven't been given, what the group $E(2)$ of isometries of $\mathbb{R}^2$ is like. What is it like?
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19 views

Norm of ideal belongs to the ideal [duplicate]

Suppose that $D$ is any number ring (i.e. $D=\mathbb Q(\alpha), \alpha \in \mathbb C$). Let $I$ be any ideal of $D$. Show that $N(I)=|D/I|$ belongs to $I$. How to start? is there a specific fact will ...
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1answer
23 views

How to prove $t(x)-x\in F_{p}$ when $p(x) \in K$, $K$ a field whose characteristic is $p>0$

Suppose $K$ is a field whose characteristic is $p>0$, and $L$ is a finite Galois extension of $K$. Also, we have an endormorphism $h$ such that $h(x)=x^p-x$ for any $x \in L$. Question: If ...
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42 views

Classify groups of order 100

So I am currently trying to Classify all groups of order 100 through an extensive proof; and this is as far as I have gotten so far, wondering how to go beyond the fact that both squares (Z4 & ...
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12 views

Find the conjugacy class of the set of symmetries of an octahedron containing a given symmetry.

Consider the following octahedron: And let G b the set of its symmetries. What is the conjugacy class of G that contains the symmetry $$(1\; 6)(2\; 4)$$?
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Algebra defined by $a^2=a,b^2=b,c^2=c,(a+b+c)^2=a+b+c$

Let $\cal A$ be the (noncommutative) unitary $\mathbb Z$-algebra defined by three generators $a,b,c$ and four relations $a^2=a,b^2=b,c^2=c,(a+b+c)^2=a+b+c$. Is it true that $ab\neq 0$ in $A$ ? This ...
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7 views

Intersection projective subspaces [on hold]

Im struggling with understanding projective spaces, I know there is a connection between projective and vectorial spaces, could someone help me showing that the intersection of projective subspaces is ...
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16 views

A commutative unital ring is a field iff its only ideals are $0$ and $R$

A commutative ring $R$ with unity is a field if and only if its ideals are $0$ and $R$. How can I prove it?
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1answer
23 views

$F$ is a field, $K$ is the splitting field of all the polynomials over $F$, how to prove that $K$ is an algebraic closure of $F$?

$F$ is a field, $K$ is the splitting field of all the polynomials over $F$, how to prove that $K$ is an algebraic closure of $F$. I know this result, but I don't know the details of the proof, which ...
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28 views

Group of polynomial $x^4+2$ in $\mathbb Q[x]$

Describe the Galois group of the polynomial $x^4+2 \in \mathbb Q[x]$. I've been able to see how to do this for $x^4-2$ and $x^4+1$ but am unsure how to do this for the polynomial above. Based on the ...
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Books on Rings without Identity

I was just wondering if anybody knows of any good books or articles that study rings (and algebras) without (or not necessarily with) identity. I have gone through Thomas Hungerford's Algebra textbook ...
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Quotient space, torus complex

I'm reading Chapter 2 of the Advances in Moduli Theory - Yuji, Kenji. My goal is the theorem of Torelli. So I find in the way of my reading the following: "...a point of the quotient space ...
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If ideal quotients of a ring are isomorphic, are these ideals isomorphic?

Suppose that $R$ is a ring, $I$ and $J$ are ideals in $R,$ and $R/I\cong R/J$ as rings. When does $I\cong J$ as $R$-modules hold?
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$H$ is a subgroup of $G$ with finite index. Prove that G has finitely many subgroups of form $xHx^{-1}$

$H$ is a subgroup of $G$ with finite index. Prove tat $G$ has finitely many subgroups of form $xHx^{-1}$. Let $h\in H$, $x\in G$ Since H is a subgroup of G $h \in G$ $\rightarrow he \in G$ ...
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1answer
41 views

13th root of 2 in field $\mathbb{F}_{13}$

Is there an easy to find $13^{th}$ root of $2$ in the field $\mathbb{F}_{13}$? I'm having trouble finding one here. Thanks!
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Almost-invariant polynomials under dihedral group action

Think about the dihedral group $D_4$ acting on the polynomial algebra $\mathbb C[x_1, \cdots, x_4]$ via generating permutations $(x_1\ x_2)$, $(x_3\ x_4)$, and $(x_1\ x_3)(x_2\ x_4)$. I'd like to ...
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2answers
49 views

Triviality of $\mathrm{Ann}(\mathfrak m)$

This question is regarding the first paragraph of the proof of Proposition 2.4 from this paper. QUESTION: Is it true that if $(0)$ is irreducible, then $\mathrm{Soc}(R)=\mathrm{Ann}(\mathfrak ...
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Some examples of "Clean topological spaces'

What is an example of a Hausdorff topological space $X$, not a singleton, such that the ring $C(X)$ of all real (or complex) valued continuous functions on $X$ is a clean ring. A clean ring is a ...
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1answer
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Factoring a homomorphism through the quotient by a normal subgroup contained in the kernel

Suppose $f\colon G\to G^{\prime}$ is a group homomorphism. Let us denote the groups additively. It is well know that such a homomorphism always can be 'factored through' the quotient ...
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Same kernels for homomorphisms of free modules

Let $f: R^n \rightarrow R^m$ be an isomorphism of free $R$-modules ($R$ commutative with unity) and $\pi_1: R^n \rightarrow R^n/\mathfrak m^n$, $\pi_2: R^m \rightarrow R^m/\mathfrak m^m$ the canonical ...
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Group ring generated by Z and the quarternion group

I want to calculate general nth power of i+j in the group ring. My idea was to find some patterns after calculating some powers of i+j, conjecture the general form of nth power of it and prove it by ...
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How does the universal quadratic functor act on morphisms?

I am trying to understand the universal quadratic functor $\Gamma$ that appears on the Whitehead exact sequence (J. Whitehead, A certain exact sequence, Annals of Mathematics 52 (1950)) particularly ...
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1answer
26 views

What is this product called?

Let $X$ be a finite set and let $2^X$ be its power set. Let $Z$ be some ring (e.g. the complex numbers; it doesn't matter). Suppose $f:2^X\to Z$ and $g:2^X\to Z$ are two functions from $2^X$ to $Z$. ...
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Problem regarding Cauchy's Theorem.

Let $G$ be a group and define $R_n(G) \subseteq G^n,R_n(G)=\{(g_1,g_2,...,g_n):g_1g_2...g_n=e\}$. $1$. Show that the map $C:G^{n-1} \to R_n(G), ...
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Product of a normal subgroup with a subgroup generated by two subgroups

Let $G$ be a group. Suppose that $N\unlhd G$ and $A,B \leq G$. I want to show that $\langle A, B\rangle N = \langle AN, BN\rangle$. Clearly, $AN \leq \langle A, B\rangle N$ and $BN \leq \langle A, ...
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1answer
74 views

For a group $G$ and subgroup $H$, is $a \sim b \iff a^{-1}b\in H$ an equivalence relation even when $H$ is not normal?

Is it true or false that defining a relation on the group $G$ based on the definition $a \sim b$ if and only if $a^{-1}b\in H$ defines an equivalence relation regardless of whether $H$ is a normal ...
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23 views

For a given non-constant polynomial $f(x)$ with integer coefficients, how many solutions are there to $f(x)\equiv 0 \mod(n)$ where $n$ is composite?

For a given non-constant polynomial $f(x)$ with integer coefficients, how many solutions are there to $f(x)\equiv 0 \mod(n)$ where $n$ is composite? Is there a general way to determine the number of ...
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18 views

Prove that $S_\infty < S_\mathbb{N}$ and $S_\infty \lhd S_\mathbb{N}$.

Let $S_\infty \subset S_\mathbb{N}$ be the set of permutations of $\mathbb{N}$ which are the identity on all but a finite number of elements. Prove that $S_\infty < S_\mathbb{N}$ and $S_\infty \lhd ...
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2answers
51 views

If $G/Z(G)$ is abelian then $G$ is abelian. [on hold]

If $G/Z(G)$ is abelian then $G$ is abelian. Give a counter example if this is not true. I know that if $G/Z(G)$ is cyclic then $G$ is abelian. $G/Z(G)$ is cyclic implies that $G/Z(G)$ is abelian but ...
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3k views

Describe all ring homomorphisms

Describe all ring homomorphisms of: a) $\mathbb{Z}$ into $\mathbb{Z}$ b) $\mathbb{Z}$ into $\mathbb{Z} \times \mathbb{Z}$ c) $\mathbb{Z} \times \mathbb{Z}$ into $\mathbb{Z}$ d) How many ...
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22 views

$\mathbb Z[r_1,r_2,…,r_n] =\mathbb Z[\frac 1m]$ [duplicate]

Question Let $r_1,r_2, ...r_n \in \mathbb Q $ Then $\mathbb Z[r_1,r_2,...,r_n] =\mathbb Z[\frac 1m]$ for some integer m $\in \mathbb Q$ I think m must be the least common multiple of the ...