A division algebra $D$ is a vector spaces over a field $F$ equipped with a bilinear product and a multiplicative neutral element $1$. All the non-zero elements of $D$ have a multiplicative inverse. Associativity is often assumed but not always. Any field is a commutative, associative division ...

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Prove that the division ring is commutative if for every $x$, $x^7=x$

I'm trying to solve a problem and I'm stuck. Here is the original problem: Let $A$ be a finite-dimensional algebra over a field $K$, such that for every $a\in A$, $a^7=a$. Show that $A$ is a ...
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1answer
42 views

Examples of division algebras

Together with the Grunwald–Wang theorem, the Albert–Brauer–Hasse–Noether theorem implies that every central simple algebra over an algebraic number field is cyclic, i.e. can be obtained by an explicit ...
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61 views

A central division algebra is not its commutator

In looking at old qualifying exam questions, I've come upon a question that has me stumped. Let $A$ be a central division algebra (of finite dimension) over a field $k$. Let $[A,A]$ be the ...
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1answer
43 views

Ideal in a certain algebra over a field

Let $K|k$ be a finite field extension. Define $D$ to be a finite dimensional $k$ division algebra. If $J$ is a nonzero two-sided ideal of $D\otimes_k K$ then by considering $K$-dimensions, I see that ...
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1answer
42 views

Show that $D \cong {\rm End}_A(D^n)$ where $D$ is a division algebra and $A\cong M_n(D)$

Define $A$ to be a finite dimensional simple algebra over a field $k$. $D$ is a $k$-division algebra (not necessarily commutative) such that $A\cong M_n(D)$ for some integer $n$. Let $L$ be a minimal ...
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1answer
27 views

Simple $M_n(D)$-module with $D$ a division ring

Define $D$ to be a division algebra over a field $k$ and $R=M_n(D)$ the $n\times n$ matrix ring over $D$. A simple $R$-module $M$ is the quotient of $R$. I can write $R=\bigoplus_j I_j$ where $I_j$ is ...
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45 views

Involutions of the second type in a division algebra

I'm trying to figure out some details about involutions of division algebra, thought maybe someone here might have a better insight. Let $k$ be a $p$-adic or number field, and let ...
2
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1answer
68 views

Ring Structures On $\mathbb {R} ^n$

In the book of Musili it is written that $\mathbb{R}^n$ is a division ring under usual addition and multiplication for $n=1,2,4$. I have understood this. But after that he said, in those cases we ...
2
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1answer
55 views

The reals as an algebra over the rationals

R, the real numbers, is an infinite dimensional commutative division algebra over the rationals Q. Is there an example of an infinite dimensional noncommutative division algebra over the rationals Q?
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1answer
24 views

Solving function in difference quotien equation

I have the problem Find the difference quotient $\frac{f(2 + h) - f(2)}{h}$ for $f(x) = \frac{1}{x^2}$. The answer they gave is $\frac{-(4 + h)}{4(2 + h)^2}$ So far I've done: $$\frac{[1/(2 + h)^2 ...
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26 views

Type of isomorphism

Let K algebraic closed field, M a simple A-module where A a K-algebra then $ End(M) \cong K $. My question is what kind of isomorphism there is between the two objects.There is a correspondence for $ ...
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1answer
27 views

finite integral domain algebras are division algebras [duplicate]

If R is a finite-dimensional algebra over a field k, and if R is also an integral domain, show that R is a division algebra over k.
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0answers
36 views

The elegant expression in terms of gcd and lcm - algebra - (2)

Definition: suppose a quantity $P$ is identified by $$ \frac{P}{k}\simeq \frac{P}{k}+1 $$ what we mean is that $$ P= 0\pmod{k}. $$ That means that when $P \to P+k$, then $$ \frac{P}{k}\to ...
2
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1answer
81 views

Endomorphisms of direct sum and division ring

How to prove that $$\operatorname{End}_R(V^{ \oplus n }) \cong M_n(D),$$ where $V$ is a simple left $R$-module and $D=\operatorname{End}_R(V)$. This is part of the proof finally leads to prove ...
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1answer
65 views

Why are there no Dual-octonions?

In the case of quaternions, we can define the traditional quaternions setting the imaginary components equal to root negative one, the hyperbolic quaternions by using root positive one, and the dual ...
1
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1answer
36 views

Example of an ordered, noncommutative division ring

Does there exist a noncommutative division ring $D$ (i. e. a field except that commutativity of multiplication is violated, e. g. the quaternions) which is also an ordered ring? Since most examples ...
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1answer
44 views

Simple $R$-module

Let $M$ be a simple $R$-module and $N=M\bigoplus M$. Then which one is true: 1) $N$ has a finite number of submodules. 2) $\operatorname{Hom}_R(N,N)$ is a division ring. 3) ...
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35 views

A proof using Fermat's Little Theorem?

Let $p$ be prime and let $a\in Z$ such that p doesn't divide a (sorry I couldn't find the symbol for it in MathJaX). Prove that if $k$ is the smallest integer such that $a^k\equiv 1 \pmod p$, then ...
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2answers
205 views

Proof: Subsequence of n integers is divisible by n?

So I'm confused and stuck on how to approach this question. Any direction in the right path would be greatly appreciated. Let $n\in N$. Prove that any sequence of $n$ integers $a_1, a_2, ... a_n$ (no ...
2
votes
1answer
41 views

direct products and direct sums of skew fields

If F is a skew field then are the arbitrary direct sums of F-modules isomorphic to their direct products (over the same index). I mean, if R is a division ring, and $\{M_i\}_{i\in I}$ some familly ...
3
votes
1answer
70 views

“Vector spaces” over a skew-field are free?

Are modules over a skew field free? That is, if $F$ is a skewfield then can any module $M$ be written as $\underset{i \in I}{\bigoplus} F$ for some indexing set $I$?
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Indefinite quaternion algebra over Q

Let $D$ be an indefinite quaternion algebra over $\Bbb Q$. We have a chosen isomorphism $\iota \colon D \otimes_{\Bbb Q} {\Bbb R} \cong {\mathrm{M}}_2({\Bbb R})$. Q: If we choose another isomorphism ...
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2answers
85 views

What can you do with octonions?

How can you calculate with them and what can you actually make up from the calculations? And what is exactly meant by normed division-algebras?
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1answer
39 views

How to check for division ring's definition.

I am rather unskilled in algebra (have never done it to be precise), but today, I had to deal with some division rings (i.e. rings with multiplicative inverse) and I came to a point where I had to ...
3
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1answer
47 views

Division ring as a $K$-algebra.

I want to solve the following question: Suppose that the division ring $\Delta$ is a $K$-algebra with $(\Delta:K)$ finite. Prove that $\Delta=K$ if $K$ is algebraically closed. Deduce that if $K$ ...
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a problem ramification index [duplicate]

Let $K$ be a finite field, $E=K((X))$, and $F=K((X^n))$, that is, $F$ is the field of formal Laurent series in $X^n$. Note that $E≅F$ as $k$-algebras by $X↦X^n$. Prove that if $L/F$ is any field ...
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1answer
47 views

A problem in division rings and Brauer group

Suppose that $D, E$ are two division rings in the Brauer group of $F$ ($Br(F)$), where $F$ is local field. Show that $D\otimes_FE$ is a division ring iff $([D:F],[E:F])=1$.
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A problem in field norm and division ring

let $D$ be a division ring with center $F$ and let $K$ be a maximal field of $D$. if $N_{D^*}(K*)$ be a maximal on $D^*$, then $K/F$ is galois.
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1answer
120 views

Prove that any integer divides zero: $a\in \mathbb Z \implies a\mid0$

Prove that any integer divides zero: $a\in \mathbb Z \implies a\mid0$ How can i prove that any integer divides zero? i tried using the definition of divisibility, but i dont know if for the formal ...
0
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1answer
51 views

Subspace of Division Algebra

I'm working on understanding the following proof: https://dl.dropboxusercontent.com/u/17606191/proof.gif but I'm having some trouble understanding some of the author's terminology. We're asked to ...
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1answer
153 views

About a proof of the Frobenius Theorem on Division Algebras

I'm trying to understand a proof of the Frobenius Theorem on Division Algebras, but my knowledge of the relevant mathematics isn't really up to scratch. The proof I'm reading is this one. I'm not ...
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2answers
76 views

There isn't any maximal finite subgroup of a multiplicative group of a division ring.

It is an exercise in division algebras and I couldn't find the answer: Suppose that $D$ is a division ring. Prove that $D^{\times}$ has no finite maximal subgroup.
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1answer
68 views

Schur's Lemma and division algebras

Let $A$ be an abelian subgroup of the unimodular group of degree $n$ (i.e. $GL(n,\mathbb Z)$). $A$ can be regarded as a group of automorphisms of a free abelian group of rank $n$ ($\mathbb Z^n$), and ...
2
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0answers
44 views

Brauer equivalence classes over $\mathbb{Q}$.

How to show that there are infinitely many Brauer equivalence classes over $\mathbb{Q}$? I proved that every such class has exactly one division algebra in it, so the question reduces to showing that ...
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0answers
59 views

A corollary to the Wedderburn-Artin theorem.

Suppose we proved the Wedderburn-Artin theorem, i.e. we have the fact that if S is a semisimple algebra over a field $F$, then $$ A \cong M_{n_1} (D_1) \times ... \times M_{n_k} (D_k), $$ where ...
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3answers
270 views

Brauer group of a field of rational numbers

Can we say anything about Brauer group of $\mathbb{Q}$? And how can we construct it?
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1answer
54 views

Factoring a polynomial over finite field $\,F_3$ that has a root

A question I am struggling with. We are asked to factor $\,f(x)=x^2+x+1$ over the field $F_3 =\{0,1,2\}$ So, I checked for a root, and I saw that $f(1) = 1^2+1+1 =0$ (because $3=0$ in $F_3$) that ...
2
votes
1answer
51 views

Is $SL_1(D)$ toplogically finitely generated, for $D$ a division algebra over a local field?

I've been struggling with this one all day, and I was wondering if someone can give me a hand with the proof. I'm not even sure if the group in question is finitely generated, so I would appreciate if ...
0
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0answers
225 views

Are the elements of a division algebra which commute with all commutators in the center of the algebra?

Let $G$ be a division algebra. If an element $c$ of $G$ commutes with every commutator of $G$, then $c$ is in the center of $G$. Is it true? If it is true, how to prove? Here commutator means an ...
0
votes
1answer
101 views

Finding Pitch Diameter of sprocket

I am currently following a tutorial on Instructables here. In the instructable to find the pitch diameter of a sprocket they use the formula on the above link. the pitch that is used is 12.70, the ...
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0answers
40 views

normed division algebra

Can we prove that every division algebra over $R$ or $C$ is a normed division algebra? Or is there any example of division algebra in which it is not possible to define a norm? Definition of normed ...
0
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2answers
293 views

Statement, equivalent to fundamental theorem of algebra

In the article THE CLASSIFICATION OF REAL DIVISION ALGEBRAS (written by R. S. PALAIS) it is said that if D is finite dimensional division algebra over $\mathbb{R}$, then: "One way of stating the ...
2
votes
1answer
140 views

Quaternion algebra and norm

Let $a \in \mathbb{Q}$ be a nonzero rational number and set $(5,a)$ and (for the associated division algebras over $\mathbb{Q}$). Let us suppose that $b$ is the norm of some element of ...
13
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1answer
197 views

what are the p-adic division algebras?

Is there a classification of division algebras over $\mathbb{Q}_p$? There are field extensions of $\mathbb{Q}_p$, but are there any others? In particular, I want to know if they are all commutative. ...
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2answers
66 views

Looking for the definition of 'locally finite-dimensional'

Recently, reading the book 'Skew Linear Groups' by M. Shirvani and B. A. F. Wehrfritz, I've encountered the following: ...
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votes
1answer
42 views

Probability : Dividing a list into 2 classes

I have a list of integer numbers ($n$). I am dividing it into two parts $n_1$ (smaller) and $n_2$ (bigger) such that the length of $n_1 \ge a*n$; $a$ is positive and $a \lt 0.5$. What is the ...
3
votes
1answer
212 views

Alternative proof of Wedderburn's little theorem

I have this exercise where I'm proving: "Every finite division ring is a field". I need only a part (c) of it: (a) show that a subalgebra of a finite dimensional central division algebra is a finite ...
0
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1answer
66 views

subalgebra of a finite dimension

Can someone please prove the following: Show that a subalgebra of a finite dimensional division algebra is a (finite dimensional) division algebra. Thanks! G.
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62 views

Additive commutators and the center of a division ring (2)

Can someone proof the following? If $D$ is a division algebra over a field $F$, then as a $Z(D)$ algebra, $D$ is generated by the additive commutators (elements of the form $xy-yx$) Thanks!
2
votes
1answer
195 views

Additive commutators and the center of a division ring

Can someone please explain me the following proof? Proposition: If $D$ is a division ring, then the division ring $R$ generated by $Z(D)$ and all additive commutators (elements of the form $xy-yx$) ...