subfield

Dec 2009
11
0
Let K be a field an let L be a finite field extension of K.
Let K be a subset of R and R a subset of L, so that for all a,b in R a+b, a*b in R.
Show: R is a subfield of L.

I still have to show the existence of inverse elements of the multiplication. The rest is all clear. But I have absolutely no real idea...

Maybe a proof by contradiction???
Can I somehow use that the field extension is algebraic?
 

NonCommAlg

MHF Hall of Honor
May 2008
2,295
1,663
Let K be a field an let L be a finite field extension of K.
Let K be a subset of R and R a subset of L, so that for all a,b in R a+b, a*b in R.
Show: R is a subfield of L.

I still have to show the existence of inverse elements of the multiplication. The rest is all clear. But I have absolutely no real idea...

Maybe a proof by contradiction???
Can I somehow use that the field extension is algebraic?
every element of R is algebraic over K. now look at the minimal polynomial of a non-zero element of R over K.
 
Dec 2009
11
0
every element of R is algebraic over K. now look at the minimal polynomial of a non-zero element of R over K.
What has this to do with multiplicatice inverse elements? The minimal polynomial of a in R is just a irreducible polynomial f in K[X] with f(a)=0 ????
 

NonCommAlg

MHF Hall of Honor
May 2008
2,295
1,663
What has this to do with multiplicatice inverse elements? The minimal polynomial of a in R is just a irreducible polynomial f in K[X] with f(a)=0 ????
let \(\displaystyle f(x)=x^n + c_1x^{n-1} + \cdots + c_{n-1}x + c_n \in K[x]\) be the minimal polynomial of \(\displaystyle 0 \neq a \in R.\) then \(\displaystyle c_n \neq 0\) and \(\displaystyle a^n + c_1a^{n-1} + \cdots + c_{n-1}a + c_n=0.\) now let \(\displaystyle b=-c_n^{-1}(a^{n-1} + c_1a^{n-2} + \cdots + c_{n-1}).\)

it's clear that \(\displaystyle b \in R\) and \(\displaystyle ab=1.\)