In mathematics, more specifically in abstract algebra, the Frobenius theorem, proved by Ferdinand Georg Frobenius in 1877, characterizes the finite dimensional associative division algebras over the real numbers. The theorem proves that the only associative division algebra which is not commutative over the real numbers are the quaternions. Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... Abstract algebra is the field of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. ... Picture of Frobenius Ferdinand Georg Frobenius (October 26, 1849 - August 3, 1917) was a German mathematician, best-known for his contributions to the theory of differential equations and to group theory. ... In mathematics, the real numbers may be described informally in several different ways. ... In mathematics, the quaternions are a non-commutative extension of the complex numbers. ...

If D is a finite dimensional division algebra over the real numbers R then one of the following cases holds

• D = R
• D = C ( complex numbers)
• D isomorphic to H (quaternions)

In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ...

Pontryagin variant

If D is a connected, locally compact division ring, then either D=R, or D=C, or D=H. Connected and disconnected subspaces of R². The space A at top is connected; the shaded space B at bottom is not. ... In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. ... In mathematics, a topological ring is a ring R which is also a topological space such that both the addition and the multiplication are continuous as maps R × R → R, where R × R carries the product topology. ...

References

• Ferdinand Georg Frobenius (1878) "Uber linear Substitutionen und bilineare Formen", Journal fur die reine und angewante Mathematik 84:1-63 (Crelle's Journal). Reprinted in Gesammelte Abhandlungen Band I, pp.343-405.
• Yuri Bahturin (1993) Basic Structures of Modern Algebra, Kluwer Acad. Pub. pp.30-2 [ISBN 0-7923-2459-5].
• R.S. Palais (1968) "The Classification of Real Division Algebras" American Mathematical Monthly 75:366-8.
• Lev Semenovich Pontryagin, Topological Groups,page 159, 1966.