Extension of Fields and Algebraically Independent Subsets.
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Let k be a subset of K be an extension of fields, and let the subset S of K be algebraically independent over k. For u belonging to K/S, show that the union of S and {u} is algebraically independent over k <=> u is transcendental over k(S).
Suggestions: For <=, prove the contrapositive. Suppose the s_1, ..., s_n belongs to S and that there is a polynomial 0 =/ f belonging to k[X_1, ..., X_n+1] with f(s_1, ..., s_n, u) = u. Use f to show that u is algebraic over k(S).
For =>, assume that (*) the union of S and {u} is algebraically independent, and suppose that u is a root of f = SUM(i -> n) a_i * X^i belonging to K(S)[X]. Argue that f = -, as follows. There are finitely many elements s_1, ..., s_m belonging to S, for which f belongs to k(s_1, ..., s_m)[X]. Now, view the coefficients a_i as rational functions of the s_j, and factor out a common denominator of the coefficients, so that f = 1/h(s_1, ..., s_m) * g (s_1,..., s_m, X) for suitable polynomials h, g. Now use the algebraic independence (*) to show that f = 0.
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Solution Summary
The proofs for the extension of fields and algebraic independence are provided in an attached Word document. The solution is rather detailed and includes a step-wise response.
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