Explore BrainMass

Explore BrainMass

    Finite Extension Field and Isomorphism

    Not what you're looking for? Search our solutions OR ask your own Custom question.

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    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

    © BrainMass Inc. brainmass.com December 24, 2021, 5:17 pm ad1c9bdddf

    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.