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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What would the units and zero divisors be?

in F3[x] for x^2-1 is x a unit or zero divisor? I was wondering what the units and zero divisors would be would they be 0,1,2,x+1,x-1,2x+2,2x-2 and then units x-2,x+2 HELP
6
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1answer
80 views

“Inverse problem” for Brauer groups

This question is really just a curiosity, but I'm really interested in the answer. Given a field $K$, we can form the set$^*$ $Br(K)$ consisting of equivalence classes of finite-dimensional central ...
2
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0answers
34 views

A question about groups generated by two elements.

Suppose a group $G=\langle a,b \rangle$ and $|G|<\infty$. How is it that the operation table for $G$ can be completely determined just by knowing $ab=b^na$ for some $n$? I don't see how this is ...
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13 views

Any transformation in $A(V)$ satisfies a polynomial of degree $2$ over $\mathbb F$.

$V$ is a two-dimensional vector space over a field $\mathbb F$; prove that every element in $A(V)$ satisfies a polynomial of degree $2$ over $\mathbb F$.
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5 views

reed muller code with parameters

Although the details of the code are not given, use the code's parameters to determine how many errors the (32, 64, 16) Reed-Muller code correct. what does this exactly mean? HELP
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6 views

Finding characteristic roots and characteristic vectors

V is a two-dimensional vector space over the field of real numbers, with a basis $v_1, v_2$. Find the characteristic roots and corresponding characteristic vectors for T defined by $v_1(T) = v_1 + ...
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15 views

Elementary Proof is isomorphism between quotient groups of gaussian integers

In order to show that, for example $Z[i]/(2-i)\cong Z/5Z$ or $Z[i]/(4-i)\cong Z/17Z$, is there any solution that explicitly constructs a homomorphism between the two sets, establishes that it is a ...
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8 views

Methods for finding Number of roots in $\mathbb{Q}_{p}$ for a polynomial

The problem is: find how many roots in $Q_{p}$ does $x^{3}+25x^{2}+x-9$ have for p=2,3,5,7. I found for $\mathbb{Z}_{p}$. Is there a way to extend from here? Also, can you suggest me a book or link ...
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1answer
14 views

how to find simple field extension $\mathbb{F_2}$?

How would I find a field extension of $\mathbb{F_2}$ that has $4$ elements? Would $x^2$ work? Since $0$ and $1$ are elements already and then $x$ and $x^2$ are units of the class? PLEASE HELP
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82 views

Do we have $(G/H)\times H \cong G$ for groups in general?

After some thought I began to suspect $(G/H)\times H \cong G$, so I tried to construct an isomorphism by hand. I came up with $\varphi: (gH, h) \mapsto gh$ which came out to work provided $G$ is ...
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1answer
32 views

Prove that $f(x)=x^2$ is a homomorphism of $\mathbb{G}$ onto $ \mathbb{H}$

Let G be an abelian group. Let $H=\{x^2 : x \in G\}$ and $K=\{x \in G: x^2=e\}$ I know I have to use the fundamental homomorphism theorem. And I also know if $f$ is a homomorphism from $G$ onto $H$ ...
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21 views

how to find congruence class modulo a polynomial

Let $\alpha$ be a root of $p(x)=x^2+x+2$ in $F3[x]$. Find: (i)$(\alpha + 1)(\alpha + 2)$ (ii) The inverse of $2\alpha + 1$ for $i$, I multiplied out and found that $$(\alpha + 1)(\alpha + ...
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1answer
14 views

General methods for solving p-adic inequality $|a^2+b|_{p}<p^{-k}$ for $a\in \mathbb{Z}$

The problem is: find integer a that satisfies the 5-adic norm inequality $|a^2+6|_{5}<5^{-4}$. I tried in vain finding it computationaly. Are there any methods from number theory to help me solve ...
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3answers
48 views

what am i misunderstanding here?

Dummit and Foote p.161 Let $G$ be an abelian group of order $n>1$. Let $n={p_1}^{a_1}\cdots {p_k}^{a_k}$ be the prime factorzation. Then, $G\cong ...
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1answer
15 views

the smallest quasigroup, which is not a group

I'm wondering, which is the smallest quasigroup, which is not a group? And how to check it?
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47 views

Field of algebraic numbers over $\mathbb{Q}$

Let $F$ be the field of algebraic numbers over $\mathbb{Q}$. I do remember that this means $F$ is a field extension of rationals over $\mathbb{Q}$. How do I show that the field extension of $F$ is ...
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2answers
54 views

Checking the maximality of an ideal

Let $R = \mathbb{Z}_{(2)}$ be the localization of $\mathbb{Z}$ at the prime ideal generated by $2$ in $\mathbb{Z}$. Then prove that the ideal generated by $(2x-1)$ is maximal in $R[x]$. Otherwise ...
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18 views

how to find complete set modulo of a polynomial

How would you go about finding the complete modulo set of $x^2-1$ in $\mathbb{F}_3[x]$? I can see $0,1,2,x+1,x-1$ are zero divisors but am a little confused on the rest.
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1answer
30 views

Isomorphisms between finite abelian groups and cyclic groups

If G is abelian of order 175 and H is cyclic of order 25 and there is a homomorphism from G onto H then what is G isomorphic to? I can see how G is isomorphic to either $C_{25} * C_7$ or to $C_5 * ...
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1answer
41 views

What would be an example such that $aH=bH$ but $Ha \neq Hb$?

Let $H$ be a subgroup of a group $G$. Assume $aH=bH$ for some $a,b\in G$ What would be an example such that $Hb\neq Ha$? I cannot imagine what would be.. Moreover, if $H$ is infinite, ...
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1answer
43 views

Isomorphisms in finite abelian groups

Let G be an abelian group of order 175 (=5*5*7). Assume $x^5=e$ has at least seven solutions. What is G isomorphic to? I see and can show that G is isomorphic to the its Sylow subgroups (orders 7 and ...
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1answer
21 views

Definition of Direct Sum of Ideals

I've been searching the internet, and I can't find a definition for the direct sum of ideals. In a previous question I posted, the author writes $M_n(D) =\oplus I_R$, where the $I_R$ are subrings and ...
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1answer
23 views

Range and kernel of groups

Let $f: G \rightarrow H$ be a homomorphism. If the range of $f$ has $n$ elements, then $x^n \in$ ker $f$ for every $x \in G$. I can kind of understand why this is true. The ker of $f$ is $\{x \in ...
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2answers
23 views

Self inverting Rings

Would it be possible for a ring to have elements that are their own additive inverses? What I mean is, would it be possible to have a ring $K$ of mathematical objects $A$ such that: $$A+A=i,\;\forall ...
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19 views

Set geometry and inclusion

Let's define first the operator for any matrix $C\in\cal M_{m,n}$,$|C|=\big(C^TC\big)^{1/2}$ where $(\cdot)^{1/2}$ is the principle square root operator. I would like to prove that the set of the ...
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1answer
33 views

Cyclic and abelian group

A group $G$ has order $25\cdot 47\cdot 17$. Is it cyclic and/or abelian? I know that a group of order $47$ or $17$ is cyclic, should I somehow use it?
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1answer
25 views

A proposition about valuation ring

Q1 $x \in m\Rightarrow x~\text{is a element of an ideal}\Rightarrow ax~\text{is a element of an ideal}\Rightarrow ax~\text{is a non-unit}$ What's the mean of $(ax)^{-1} \in B$ ? Q2 $x,y \in ...
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1answer
21 views

Normal subgroup $N$, subgroup $U$, then $UN/N = U/N$.

Let $G$ be a group and $N \unlhd G$ a normal subgroup, $U \le G$ some subgroup. Then I guess $U / N$ is always some group, and moreover $U / N = UN / N$, because $UN / N = \{ unN : u \in U, n \in N \} ...
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55 views

Prove $M$ is a Maximal Ideal in $\Bbb Z\times \Bbb Z$

A problem from introduction to abstract algebra by Hungerford. It asks: If $p$ is a prime integer, prove that $M$ is a maximal ideal in $\mathbb Z \times \mathbb Z$, where $M =\{(pa,b)\mid a,b\in ...
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49 views

valuation ring is a field?

suppose $a$ and $a'$ are units of $B$ ,$b$ and $b'$ are the elements of any ideal of $B$. $x$ is a element of $K$. $K$ consist of $a/a,a/b,b/a,b/b$ $\color{green} x=a/a' \Rightarrow x\in ...
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0answers
37 views

Prove that a(mn)=a(m)a(n), (n,m)=1

Given a positive integer $n$ where $a(n)$ is the number of non-isomorphic abelian groups of order n. 1) Prove that $a(mn)=a(m)a(n), (n,m)=1$ 2) Prove that $a(p^k)$ is the number of partitions of k, ...
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1answer
26 views

Galois group of the splitting field of $x^3-2$

I want to find $Gal(E/\mathbb{Q})$ where $E$ is the splitting field of $f(x)=x^3-2$. I started out finding the zeros, which is $2^{1/3},2^{1/3}\omega, 2^{1/3}\omega^2 $, where ...
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0answers
17 views

Rigidity for Lie Groups

This may be a very dumb question but I was wondering if the following train of logic is correct: We know a connected Lie group $G$ is isomorphic to the quotient $G\cong \tilde{G}/\Gamma$ where ...
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1answer
22 views

A question about subexponential growth about group

Recall a group $\Gamma$ is said to have subexponential growth if lim sup$|E^{n}|^{1/n}=1$ for every finite subset $E\subset \Gamma. (E^{n}=\{g_{1}g_{2}...g_{n}: g_{i}\in E\}.)$ My question is: Can we ...
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1answer
60 views

Introducing multiplication of cosets

So, i have encountered two ways to introduce the multiplication of cosets, and i want to understand exactly what is happening in each, specifically in light of the multiplication of cosets being ...
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145 views
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Putting down axioms for some symbols. Playing with their consequences qualitatively and symbolically. Building theories. The book?

I am interested in the design and building of theories. By building theories, I mean putting down axioms of various kinds, over various fields, exploring their perhaps interesting, or probably boring, ...
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1answer
29 views

Query on homomorphism.

If we say that $H:A\rightarrow B$ is a homomorphism from A to B, does it mean that A is homomorphic to B or B is homomorphic to A?. Are the two statements actually different? What is meant by the ...
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1answer
37 views

Proof of Wedderburn's Theorem

I've been going through a proof of Wedderburn's theorem: and I'm stuck on the very last part, where the author refers to example 2.1.4. (linked below). I don't understand what $D^n$ means, or why it ...
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0answers
28 views

Let $G$ and $H$ be finite groups and $(g,h) \in G \bigoplus H$. Condition for $\langle g,h \rangle = \langle g \rangle \bigoplus \langle h \rangle ?$

Let $G$ and $H$ be finite groups and $(g,h) \in G \bigoplus H$. State a necessary and sufficient condition for $\langle g,h \rangle = \langle g \rangle \bigoplus \langle h \rangle$ Attempt : Let $l = ...
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1answer
51 views

Surjective homomorphism on Commutative Ring

Let $A$ be a commutative ring, $R= A[x_{1},...,x_{n}]$ and $(a_{1},...a_{n}) \in A^{n}$ . Let $\phi : R \to A$ be defined by $\phi (f(x_{1},...,x_{n}) = f(a_{1},...,a_{n})$. Then show that $\phi$ is a ...
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Is every subgroup of the product of two cyclic groups is again a product of two cyclic groups?

Well, this is my question. Is every subgroup of the product of two cyclic groups is again a product of two cyclic groups (maybe one being trivial)? Thanks!
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1answer
32 views

$4|(p-1) \implies$ there is an element $x$ of order $ 4$ modulo $p$.?

"$p \equiv 1 \mod 4 \implies 4 \mid (p-1) \implies$ there is an element $x$ of order $4$ modulo $p$." I am having a difficult time understanding why this implies there is an element $x$ of order $4$. ...
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0answers
20 views

Variety satisfying an identity.

V is a variety of commutative semi group satisfying the identity $x^2 = x^3$. I need to prove that: $|F_V(\{x_1\dots,x_n\})|$ = $3^n -1$. Any hints on this ? $F_V$ is V-free algebra.
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1answer
28 views

How does the base of a group determine the “sort” of the elements in the group

I'm trying to study groups in Mathematica, and I've asked a question on Mathematica.SE that perhaps only someone from Math.SE could answer. Related: How does ...
5
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1answer
89 views

When are generalized Severi-Brauer varieties trivial?

Let $F$ be a field and $A$ be an $F$-central simple algebra of degree $n$. Let $0< k< n$ and let $SB_k(A)$ denote the generalized Severi-Brauer variety: if $E/F$ is a field extension, ...
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Direct proof for the independence of $\operatorname{Tor}$

It is known that $\operatorname{Tor}$ is independent of the choice of the resolution. More specifically, I am trying to do the exercise 1 (c) of Vick's homology theory. The author gives the ...
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2answers
61 views

What is the reason for stating Cayley's theorem this way?

In my notes, Cayley's theorem reads: Any group $G$ is isomorphic to a subgroup of $\text{Sym}\, X$ for some $X$. On the other hand, several sources (such as Wikipedia) give a slightly more ...
2
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2answers
48 views

(12345) is an even permutation of S_5. True or False?

The answer i had for this question was True, yet i'm not sure. Well, from what I know so far was that: $(12345)$ can be expressed as a number of 4 transpositions such as: $(12)(23)(34)(45)$ which is ...
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1answer
31 views

Can we replace the $B$ to $A$ in this proposition

I am working through Atiyah's Commutative algebra and am having question with the following proposition: $\text{Page 63:}$ Proposition 5.15. Let $A$ $\subseteq$ $B$ be integral domains, $A$ ...
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2answers
23 views

What the difference between $A/m$ and $A_0$

$A$ is a integral domain. $m$ is a maximal ideal . $A_0$ is the localization of $A$ by $A-0$.(Field of fractions) $A/m$ is the quotient at $m$. What the difference between $A/m$ and $A_0$?