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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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    https://brainmass.com/math/basic-algebra/finite-extension-field-and-isomorphism-39353

    Solution Preview

    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

    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.

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