Advanced Algebra II
Homework 1 due on Feb. 27, 2004
In this homework, we assume that char(K) = p 6= 0 unless otherwise stated.
(1) Give an example of finite extension F/K which is neither sep- arable nor purely inseparable.
(2) Give an example of finite extension which is not a simple ex- tension.
(3) If a ∈ K − Kp, then xpn − a is irreducible for every n ≥ 1.
(4) Let F/K be an extension. Show that u ∈ F is separable over K if and only if K(u) = K(upn) for all n ≥ 1.
(5) Let F/K be an algebraic extension. Suppose that u ∈ S, v ∈ P . Show that K(u, v) = K(u + v).
(6) Examine the property of being purely inseparable under ”ex- tension”,”lifting”, and ”compositum”.
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