Math 201C Homework
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چکیده
i=0 ai,kv i = 0. Recall that, since v is trancendental over K, the set {1, v, v, ...} is linearly independent over K. This implies that ai,k = 0 for all k > 0 (otherwise there would be a finite linear combination of powers of v which is zero, hence v would be algebraic over K). Thus each hi above is of the form hi(x) = ai,0 (a constant polynomial), namely hi ∈ K. But then (2) implies that v is algebraic over K, a contradiction unless each ai,0 = 0. However, this is also a contradiction since ai,k = 0 for all i, k implies that η is the zero polynomial. Thus ψ ∈ K(v)[x] is a nonconstant polynomial such that ψ(u) = 0. Therefore u is algebraic over K(v). Exercise 5.1.8. If u ∈ F is algebraic of odd degree over K, the so is u and K(u) = K(u). Proof.
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تاریخ انتشار 2011