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 k_p 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)