Argue that every finite extenstion field of R is either R itself or is isomorphic to C.
Note: R is set of all real numbers
C is set of all complex numbers
Definitions (from Fraleigh)
If an extension field E of a field F is of finite dimension n as a vector space over F, then E is a finite extension of degree n over F.
Well certainly C is a finite extension of degree 2 over R, since C is the vector field R(i) and all elements of C are solutions to equations of degree 2 in R. And R is a finite extension of degree 1 over itself.
Theorem 7.12 from Fraleigh: A finite extension field E of a field F is an algebraic extension of F
Algebraic extension: every element of E is algebraic over F.
Algebraic over F: a is algebraic over F if f(a) = 0 for some nonzero polynomial f(x) in ...
Finite Extension Fields and Isomorphisms are investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.