# subfield

• May 31st 2010, 05:47 AM
Icarus0
subfield
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...

Can I somehow use that the field extension is algebraic?
• May 31st 2010, 06:01 AM
NonCommAlg
Quote:

Originally Posted by Icarus0
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...

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.
• May 31st 2010, 08:46 AM
Icarus0
Quote:

Originally Posted by NonCommAlg
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 ????
• May 31st 2010, 04:54 PM
NonCommAlg
Quote:

Originally Posted by Icarus0
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 $f(x)=x^n + c_1x^{n-1} + \cdots + c_{n-1}x + c_n \in K[x]$ be the minimal polynomial of $0 \neq a \in R.$ then $c_n \neq 0$ and $a^n + c_1a^{n-1} + \cdots + c_{n-1}a + c_n=0.$ now let $b=-c_n^{-1}(a^{n-1} + c_1a^{n-2} + \cdots + c_{n-1}).$

it's clear that $b \in R$ and $ab=1.$