实封闭域上的代数

Algebras over Real Closed Field

在建立了实封闭域F上复元素域C与四元素体H后得到了:(1)全阵代数F2n中有子代数同构于C,全阵代数F4n中有子代数同构于H;(2)F上代数扩张体只有F、C和H;(3)设F是域K里上维数有限的真子域,则F是实封闭的K是代数闭域且K=F((-1)～(1/2));(4)设A是F上的有限维代数,①若A是可除代数,则A同构于F、C或H,②若A是中心可除代数,则A同构于F或H,③若A是单代数,则A同构于全阵代数Fn、Cn与Hn中之一,④若A是中心单代数,则A同构于全阵代数Fn或Hn,⑤若A没有非零幂零理想,则A=sum Mni from i=1 to l,其中Mni∈{Fni,Cni,Hni},i=1,2,…,l。

The author constructs complex element field and quartic element division ring over the real closed filed.The following results are obtained： 1.Let F be a field,then（1） there is a subalgebras in the total matrix algebra F2n which is isomorphic to the complex element field C over F（C=F（i））,（2） there is a subalgebras in the total matrix algebra F4n which is isomorphic to the quartic element division ring H over F（H=F（1,i,j,k））.2.The only algebraic extension division rings over the real closed field F are（1） F,（2） C,and（3）H.3.Let F be a proper subfield of finite codimensional in the field K（∞）.Then F is real closed if and only if Kis algebraically closed and K=F（（-1）～（1/2））.4.Let A be a finite dimensional algebra over the real closed field F,then（1） up to isomorphism the only division algebras A are F,C or H,（2） up to isomorphism the only central division algebras A are F or H.（3） up to isomorphism the only simple algebras A are Fn,Cn or Hn.（4） up to isomorphism the only central simple algebras A are Fn or Hn.（5） If A has no non-zero nilpotent ideals,then A is a direct sum of finitely many total matric algebras Mni over F,where Mni∈{Fni,Cni,Hni}.