Talk:Perfect field

Perfect closure?
The article contained the paragraph
 * The first condition says that, in characteristic p, a field adjoined with all p-th roots (usually denoted by $$k^{p^{-\infty}}$$) is perfect; it is called the perfect closure, denoted by $$k_p$$. Equivalently, the perfect closure is a maximal purely inseparable subextension. [Commented out: Let $$E/k$$ be an algebraic extension, and $$k_s$$ separable closure (a maximal separable subextension). While $$E/k_s$$ and $$k_p / k$$ are purely inseparable, $$E/k_p$$ need not be separable. On the other hands, one has:] If $$E/k$$ is a normal finite extension, then $$E \simeq k_p \otimes_k k_s$$.[Reference:Cohn, Theorem 11.4.10]

The second part doesn't make sense to me. "Equivalently, the perfect closure is a maximal purely inseparable subextension." A subextension of what? All we have at this stage is a field k. Maybe a maximal purely inseparable subextension of the algebraic closure of k? The formula $$E \simeq k_p \otimes_k k_s$$ cannot possibly be true: E=k with k an imperfect field is a counter example, because kp is bigger than k in that case.

I removed everything beginning with "Equivalently...". Please restore if it can be clarified. AxelBoldt (talk) 20:28, 15 December 2011 (UTC)


 * It's actually not incorrect; you have to interpret in a right way. The problem is the paragraph doesn't make a careful distinction of absolute closure and relative closure. -- Taku (talk) 15:18, 11 March 2012 (UTC)

reals
It would be nice to reference the reals as perfect [or not].Chris2crawford (talk) 12:26, 1 October 2015 (UTC)