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Basis and Finite and Infinite Field Extensions

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Find a basis for the extension of and also calculate
We know already that is infinite.
Give an example of fields and (with neither nor equal to ) such that:
a) and are both infinite
b) is infinite and is finite.

(See attached file for full problem description with proper symbols)

All gaps are Q, the rational numbers.

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1. Find a basis of the extension of and also calculate
Let , then , ...

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Basis and Finite and Infinite Field Extensions are investigated. The solution is detailed and well presented.

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