Frobenius theorem (real division algebras)

The Frobenius theorem, 1877 by Ferdinand Georg Frobenius proved part of the mathematical branch of algebra . The theorem states that apart from isomorphism there are only three finite-dimensional, associative division algebras over the real numbers : themselves, the complex numbers and the quaternions . ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {C} }$ ${\displaystyle \mathbb {H} }$

The theorem was independently proven in 1881 by Charles Sanders Peirce . The theorem restricts the existence of associative division algebras over the real numbers to the dimensions 1, 2 and 4. The inventor of the quaternions, William Rowan Hamilton , had long searched for such an algebra in three dimensions, which is excluded by the theorem. If one drops the condition of associativity and demands commutativity, Heinz Hopf proved in 1940 that the corresponding finite-dimensional division algebras have a maximum of dimension 2. There is topological proof that there are division algebras over the real numbers only for n = 1, 2, 4, 8 (see division algebra ). ${\displaystyle \mathbb {R} }$

Of these three division algebras, only the quaternions are an oblique field with a non-commutative multiplication. Since and are the only finite-dimensional, commutative and associative division algebras over the real numbers, it must be shown for the proof of Frobenius' theorem that the quaternions form the only finite-dimensional non-commutative skew body over : ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {C} }$ ${\displaystyle \mathbb {R} }$

Let be a finite dimensional non-commutative skew body over . Then there is an -algebra isomorphism . ${\displaystyle D}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {H} \rightarrow D}$

proof

${\displaystyle \mathbb {C} }$except for isomorphism, is the only finite real field extension of . so there is no oblique body over and it holds ( denotes the center of ). ${\displaystyle \mathbb {R} }$ ${\displaystyle D}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle \mathbb {R} =Z(D)}$ ${\displaystyle Z(D)}$ ${\displaystyle D}$

Consequently contains a maximum part of the body with ${\displaystyle D}$ ${\displaystyle \mathbb {C} '}$

${\displaystyle (\dim _{\mathbb {R} }\mathbb {C} ')^{2}=\dim _{\mathbb {R} }D.}$

Since is non-commutative, and . ${\displaystyle D}$ ${\displaystyle \mathbb {C} '\simeq \mathbb {C} }$ ${\displaystyle \dim _{\mathbb {R} }D=4}$

It now ranges, one - Algebrenhomomorphismus specify, for the injectivity then follows as a simple ring , and the surjectivity follows dimensional reasons. ${\displaystyle \mathbb {R} }$ ${\displaystyle \varphi \colon \mathbb {H} \rightarrow D}$ ${\displaystyle \mathbb {H} }$

It is with and Galois over with Galois group , where ${\displaystyle \mathbb {C} '=\mathbb {R} \oplus j\mathbb {R} }$ ${\displaystyle j^{2}=-1}$ ${\displaystyle \mathbb {C} '}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \{id,\sigma \}}$

${\displaystyle \sigma \colon \mathbb {C} '\rightarrow \mathbb {C} ',~a+bj\mapsto a-bj.}$

According to Skolem-Noether's theorem, there is now such that . Now applies: ${\displaystyle u\in D\backslash \{0\}}$ ${\displaystyle \forall x\in \mathbb {C} '\colon \,\sigma (x)=uxu^{-1}}$

• ${\displaystyle u^{2}\in \mathbb {R} }$. Proof: It is , or . So follow and . Since Galois is over , follows . ${\displaystyle u^{2}ju^{-2}=\sigma ^{2}(j)=j}$ ${\displaystyle u^{2}j=ju^{2}}$ ${\displaystyle u^{2}\in Z_{D}(\mathbb {C} ')=\mathbb {C} '}$ ${\displaystyle \sigma (u^{2})=uu^{2}u^{-1}=u^{2}}$ ${\displaystyle \mathbb {C} '}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle u^{2}\in \mathbb {R} }$

• ${\displaystyle u^{2}<0}$. Proof: Adopted . Then and due also . Contradiction. ${\displaystyle u^{2}>0}$ ${\displaystyle u\in \mathbb {R} }$ ${\displaystyle \mathbb {R} =Z(D)}$ ${\displaystyle -j=\sigma (j)=uju^{-1}=j}$

So we get a representation with . Our sought algebra homomorphism is now induced by ${\displaystyle u^{2}=-r^{2}}$ ${\displaystyle r\in \mathbb {R} ^{+}}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \varphi \colon \mathbb {H} \rightarrow D}$

${\displaystyle {\begin{pmatrix}1&0\\0&1\end{pmatrix}}\mapsto 1,\quad {\begin{pmatrix}0&1\\-1&0\end{pmatrix}}\mapsto j,\quad {\begin{pmatrix}i&0\\0&-i\end{pmatrix}}\mapsto ur^{-1},\quad {\begin{pmatrix}0&i\\i&0\end{pmatrix}}\mapsto ur^{-1}j,}$

because it applies . ${\displaystyle -j=\sigma (j)=uju^{-1},~uj=-ju}$

By looking at the appropriate group tables , the claim follows.

See also

• Gelfand-Mazur theorem

literature

• M. Koecher , R. Remmert : Isomorphy theorems from Frobenius and Hopf. In: H.-D. Ebbinghaus among others: Numbers. Springer Verlag, 1983.
• Ina Kersten : Brewer groups. Universitätsdrucke Göttingen, Göttingen 2007, pp. 52–54, PDF (accessed on July 18, 2016).

Individual evidence

1. ↑ Frobenius: About linear substitutions and bilinear forms. In: J. Reine Angew. Math. Volume 84, 1877, pp. 1–63, SUB Göttingen , reprinted in Frobenius: Gesammelte Abhandlungen. Volume 1, pp. 343-405.
2. ↑ Appendix from CS Peirce on Benjamin Peirce : Linear associative algebras. In: American Journal of Mathematics. Volume 4, 1881, pp. 221-226.
3. ↑ A proof of the theorem can be found, for example, in M. Koecher, R. Remmert, Chapter 7, in: Ebbinghaus et al.: Numbers. Springer 1983.
4. ↑ An elementary proof comes from Richard Palais : The classification of real division algebras. In: American Mathematical Monthly. Volume 75, 1968, pp. 366-368.
5. ^ Ina Kersten: Brewer groups. P. 38.
6. ^ According to Ina Kersten: Brewer groups. See literature .