When reading about field theory the only example of a skew field I have ever come across are the quaternions. Are there other examples?
The topic of division rings is very interesting and there are many deep results and construction around it.
Roughly speaking there are two kinds of division rings:
-The ones that are finite dimensional over its center (e. g. the quaternions). Here one has to point out that there is a famous theorem of Wedderburn stating that any finite division ring is commutative. On the other hand one of the goals of "the Brauer group" is to study finite dimensional division rings with a given center k. This is a very classical topic, that was one of the starting points of the non-commutative algebra and some of the fundamental results are due to E. Noether. A very interesting computation is the Brauer group of the rationals which shows that one can construct infinitely many non-isomorphic finite dimensional division rings with center Q that are not isomorphic to quaternion algebras.
- Forgetting about the finite dimensionality, one has a theory of field of fractions for non necessarily commutative domains that produces construction of division rings. In the non commutative setting it is not true that any domain has a field of fractions. The naive approach of construct a field of fractions copying the commutative setting was followed by Ore already in the thirties, the outcome was the theory of Ore domains. Nice examples of Ore domains are the so called skew-polynomial rings, that is a ring of polynomials such that the coefficients and the variable commute up to some automorphism and/or derivation. For example, if C denotes the complex numbers and \alpha is the complex conjugation the skew-polynomial ring R=C[x,\alpha] is a ring with elements the polynomials of the form z_0+\dots +z_nx^n , z_i in C (coefficients always on the left!) and with the rule xz=\alpha (z)x for any z\in C. R is an Ore domain and this means that has a field of fractions which is going to be a non-commutative one!
This is really a wide topic, I think a good source for some constructions and examples (and to have a better introduction to the subject than the one I am doing here!) is the book by T. Y. Lam
A First Course in Noncommutative Rings, Graduate Texts in Mathematics, Vol. 131, Springer-Verlag, 1991 (Second Edition, 2001).
It is impossible to talk about division rings without mentioning P.M. Cohn who developed a complete theory of localization that, between other things, solved the so called Artin problem: There is a division ring extension D subset F such that the dimension of F as a left D vector space does not coincide with the dimension of F as a right D vector space. Cohn has written many interesting books (e.g. Skew Fields, for the Encyclopedia of Mathematics)
I know only the quaternion Division rings ,for more see http://www.encyclopediaofmath.org/index.php/Skew-field
The topic of division rings is very interesting and there are many deep results and construction around it.
Roughly speaking there are two kinds of division rings:
-The ones that are finite dimensional over its center (e. g. the quaternions). Here one has to point out that there is a famous theorem of Wedderburn stating that any finite division ring is commutative. On the other hand one of the goals of "the Brauer group" is to study finite dimensional division rings with a given center k. This is a very classical topic, that was one of the starting points of the non-commutative algebra and some of the fundamental results are due to E. Noether. A very interesting computation is the Brauer group of the rationals which shows that one can construct infinitely many non-isomorphic finite dimensional division rings with center Q that are not isomorphic to quaternion algebras.
- Forgetting about the finite dimensionality, one has a theory of field of fractions for non necessarily commutative domains that produces construction of division rings. In the non commutative setting it is not true that any domain has a field of fractions. The naive approach of construct a field of fractions copying the commutative setting was followed by Ore already in the thirties, the outcome was the theory of Ore domains. Nice examples of Ore domains are the so called skew-polynomial rings, that is a ring of polynomials such that the coefficients and the variable commute up to some automorphism and/or derivation. For example, if C denotes the complex numbers and \alpha is the complex conjugation the skew-polynomial ring R=C[x,\alpha] is a ring with elements the polynomials of the form z_0+\dots +z_nx^n , z_i in C (coefficients always on the left!) and with the rule xz=\alpha (z)x for any z\in C. R is an Ore domain and this means that has a field of fractions which is going to be a non-commutative one!
This is really a wide topic, I think a good source for some constructions and examples (and to have a better introduction to the subject than the one I am doing here!) is the book by T. Y. Lam
A First Course in Noncommutative Rings, Graduate Texts in Mathematics, Vol. 131, Springer-Verlag, 1991 (Second Edition, 2001).
It is impossible to talk about division rings without mentioning P.M. Cohn who developed a complete theory of localization that, between other things, solved the so called Artin problem: There is a division ring extension D subset F such that the dimension of F as a left D vector space does not coincide with the dimension of F as a right D vector space. Cohn has written many interesting books (e.g. Skew Fields, for the Encyclopedia of Mathematics)
For any simple module M over any ring R, the ring of R-endomorphisms of M is a division ring. This produces a LOT of examples :)
Dolors Herbera pointed out that "P.M. Cohn [...] developed a complete theory of localization that, between other things, solved the so called Artin problem: There is a division ring extension D subset F such that the dimension of F as a left D vector space does not coincide with the dimension of F as a right D vector space."
An entertainingly concrete alternative way to state this (if I remember Cohn's wittily titled _Free Rings and Their Relations_ correctly), there is an m-by-n matrix over D with m not equal to n (he gives an example with (m,n)=(2,3), I believe) that is both left-invertible and right-invertible. Thus one's intuitions about "rank" of a matrix, founded as they are likely to be--unless one is a full-on algebraist, rather than, e.g., an analyst or differential geometer or topologist on holiday in a foreign field--on experience with matrices over the real or complex numbers, don't survive too well in the wilder world of rings that are non-commutative (or worse). ... Actually, you can "give" an example yourself (but you need to have Cohn's machinery to show that it exists as a ring): just write down an m-by-n matrix A, an m-by-m matrix B, and an n-by-n matrix C, all with "generic" entries a_{i,j}, b_{p,q}, c_{r,s} (in a "free ring with unity"), for a total of mn+m^2+n^2 variables; then write down the product matrices BA and AC, and set them equal to the m-by-m and n-by-n identity matrices, respectively, giving a total of m^2+n^2 relations on those variables; finally, declare that D is the division ring generated by those variables subject to (just) those relations (and their consequences). CAUTION: the variables ARE NOT COMMUTATIVE. (I can't remember if they're associative, but I suppose so.) Problem to puzzle over: why can't you do this if one of m, n equals 1?
Artin's problem on dimensions of extensions of division rings is much more subtle than the one of finding "invertible non square matrices" commented on Lee Rudolph's answer.
A typical source of examples for such phenomena is the ring of endomorphism ring of an infinite dimensional vector space.
Let k be a field, and let V be a countable dimensional k-vector space with basis e_0,e_1,e_2......
Let a\colon V\to V be the k-linear map determined by a(e_i)=e_2i and let b\colon V\to V be the k-linear map determined by b(e_i)=e_{2i+1}. It is not difficult to find c, d elements of R=\End _k(V) such that (a,b)\left(c//d \right)=1 and the multiplication in the opposite order is a 2\times 2 identity matrix.
Some general remarks:
Assume that R is an associative ring, with 1\neq 0 (i.e. a nonzero ring) and non necessarily commutative.
As in linear algebra, an m\times n matrix A gives, by left multiplication, and endomorphism of right R-modules f\colon R^n\to R^m. The matrix A is invertible if and only if f is an isomorphism.
If R is commutative, f isomorphism implies n=m. This is because if M is a maximal ideal of R, then tensoring by R/M gives an isomorphism between the finite dimensional R/M-vector spaces (R/M)^n and (R/M)^m. Hence, n=m.
For a non commutative ring not only we do not have such handy way to associate a finitely dimensional vector space to a free module but, as shown in the example, the conclusion is false. As it happens with interesting concepts we then get a name: rings with IBN (Invariant Basis Number), which are the rings such that R^n\cong R^m as right modules, for n and m finite, implies n=m. Translating the concept to matrices one sees that this is a symmetric definition (i.e. it is also true for finitely generated free left R-modules).
Skew fields and, more generally, rings embeddable in skew fields have IBN. So Artin's problem for skew fields is really a different matter.
Let me add one detail to the very nice reply of Dolors Herbera.
For every field F there is a non-commutative division ring R such that
F is the center of R and R is infinite-dimensional over F.
This is Proposition 2.3.5 from Cohen's book.
@Dolors: You write that there are infinitely many non-isomorphic finite dimensional division rings with center Q that are not isomorphic to quaternion algebras. Could you give a reference for this fact?
This is a consequence of the computation of the Brauer group of the rationals. I think that a very classical reference for that is
Pierce, Richard (1982). Associative algebras. Graduate Texts in Mathematics 88.
Now I am just writting what I remember, If it does not work I can check it better.
For having just a quick overview on the Brauer group the wikipedia article
http://en.wikipedia.org/wiki/Brauer_group
seems OK
I think that another example can be the so-called transfer functions of nonlinear systems.
Assume we have a nonlinear system descirbed by a nonlinear differential equation
y(n) = f(y,...,y(n-1),u,...,u(m))
where f is meromorphic. Then, after differentiating it, we can write
dy(n) + an-1 dy(n-1) + ... + a0 dy = bm du(m) + ... + b0 du
for some functions ai, bj. The two associated polynomials
a(s) = sn + an-1 sn-1 + ... + a0
b(s) = bm sm + bm-1 sm-1 + ... + b0
are from a (non-commutative) skew polynomial ring. Elements of the associated skew field of left fractions
F(s) = a-1(s) * b(s)
are the so-called transfer functions of nonlinear systems, as we have
dy = F(s) du
I just saw this discussion. As far as it is still relevant, I have described a most simple example of a skew field extension with finite but different left and right degree, namely left degree 3 and right degree 2. It can be found in
Proposition 3.2(b) and its proof in:
Treur, J., Polynomial extensions of skew fields. Journal of Pure and Applied Algebra 67( 1), 1990, 73-93.
https://www-sciencedirect-com.vu-nl.idm.oclc.org/science/article/pii/002240499090164D
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