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?