**3**

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**1**answer

203 views

### Division ring on a field

Suppose that $F$ is a field. Show that there exists a $F$-division algebra $D$ with two elements $a\neq b\in D$ such that $a^2-2ab+b^2=0$.
In the field extensions we know that $a^2-2ab+b^2=0$ if and ...

**1**

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**0**answers

78 views

### Galois extensions inside a division ring

Let $D$ be a division ring which has finite dimension over its centre.
Q1. Under which conditions can one find a maximal subfield $K$ of $D$ and a proper subfield $L$ of $K$ such that $K/L$ is Galois?...

**1**

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**1**answer

74 views

### For a division ring $D$, does $[D:C_D(a)]_{right}$ vary when $D$ is enlarged?

In a commutative field $K$, the Zariski dimension of an algebraic subset of $K^n$ over $K$ does not vary if one enlarges $K$ if I understood well. In particular, for two Zariski-closed vector spaces $...

**2**

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**0**answers

76 views

### If $A$ is an integer ring such that each $P \in A_L[X]$ has a finite number of zeros in $A$, is $A$ commutative?

Let $A$ be a ring in which the product of any two nonzero elements is nonzero (we shall say that $A$ is an integral domain, even if $A$ is non commutative). It is well-known that if $A$ is commutative,...

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**1**answer

91 views

### Semisimple elements in division algebras

I found the following exercise at page 85 of the Strade-Farnsteiner's book "Modular Lie algebras and their representation": Let $D$ be a finite-dimensional division ring over a field $F$ of ...

**1**

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**1**answer

248 views

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

I asked this quetion five days ago at http://math.stackexchange.com/questions/406669/are-the-elements-of-a-division-algebra-which-commute-with-all-commutators-in-the Some good people have given good ...

**6**

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**1**answer

349 views

### Does there exist an infinite non-commutative division ring with finite center?

Does there exist an infinite non-commutative division ring with finite center?

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**2**answers

437 views

### What structure supports division to a unique quotient and remainder?

This has been bugging me for a while.
According to https://en.wikipedia.org/wiki/Euclidean_division, if I divide integer $a$ by integer $b$, I get unique $t$, $r$ such that $a = t b + r$, $0 \le r &...

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**1**answer

331 views

### Analysis in Division Rings

In the question here, the subject of "Analysis in Positive Characteristic" is mentioned. Looking at Wikipedia's local field, this is the final type of analysis in local fields to be developed ...

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**1**answer

218 views

### (Non-)existence of skew fields satisfying a SGPI (=skew generalized polynomial identity)

Let $K$ be a skew-field, infinite dimensional over its center $F$.
From Kaplansky's PI-theorem it then follows that $K$ cannot satisfy a polynomial identity (the theorem says that primitive PI-...

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**2**answers

436 views

### Bimodules over division rings

Inspired by other questions i have two questions about modules over division rings: given a division ring $D$ with center $Z(D)=K$. One has the notion of dimension for left modules (vector spaces) $V$ ...

**0**

votes

**1**answer

287 views

### Uniqueness of maximal subfields

Let D be a division ring with center Z. Let R and K be two maximal subfields of D, both purely inseparable of exponent one ( means the p power of each of them in Z).
Why are R and K isomorphic?
Or a ...

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votes

**2**answers

726 views

### units in distinct division algebras over number fields---are they definitely not isomorphic as abstract groups?

This is really an irrelevant question in the sense that the answer isn't remotely "logically crucial for the Langlands programme" or whatever---it's just something that occurred to me when writing ...

**14**

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**4**answers

1k views

### Dimension of central simple algebra over a global field “built using class field theory”.

If $F$ is a global field then a standard exact sequence relating the Brauer groups of $F$ and its completions is the following:
$$0\to Br(F)\to\oplus_v Br(F_v)\to\mathbf{Q}/\mathbf{Z}\to 0.$$
The ...

**10**

votes

**2**answers

822 views

### Free division rings?

Does it make sense to talk about, say, the free division ring on 2 generators? If so, does the free division ring on countably many generators embed into the free division ring on two generators?

**6**

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**5**answers

1k views

### Proof a Weyl Algebra isn't isomorphic to a matrix ring over a division ring

Can anyone prove that a Weyl Algebra is not isomorphic to a matrix ring over a division ring?