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Finite Extension Field and Isomorphism

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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

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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 ...

Solution Summary

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