Fields Extensions/Algebraic
Let F be an extension field of K. Clearly F is a vector space over K. Let u be an element of F. Show that the subspace spanned by {1, u, u^2, ...} is a field IF and ONLY IF (iff) u is algebraic over K.
Let S be the subspace of F.
Hint for the "-->" of proof. If S is a field and u is not equal to 0, then 1/u is in S. What does that imply?
Hint for the "<---" Show S is closed under +, -, and x. Then need to show that if v is an element of S with v not equal to 0, then 1/v is an element of S. Suppose u is algebraic over K with minimal polynomial p(x) then consider how multiplicative inverses are found in K[x]/
This solution is comprised of a detailed explanation to show that the subspace spanned by {1, u, u^2, ...} is a field IF and ONLY IF (iff) u is algebraic over K.
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