My first idea is using Baire category theorem since I thought an infinite algebraic extension should be of countable degree. However, this is wrong, according to [this post](https://math.stackexchange.com/questions/1293807/is-the-degree-of-an-infinite-algebraic-extensions-always-countable).

This approach may still work if it is true that **infinite algebraic extensions of complete fields have countable degree**. For instance, infinite algebraic extensions of local fields are of countable degree. I wonder if this is true for general case.

Another possible approach is to generalize the proof in [Bosch, Güntzer, Remmert: Non-Archimedean Analysis](http://www.springer.com/us/book/9783540125464), Lemma 1, Section 3.4.3. Where they use [Krasner's Lemma](http://en.wikipedia.org/wiki/Krasner%27s_lemma) to prove that if the algebraic closure of a complete field is of infinite degree then it is not complete. However, Krasner's Lemma only works for separable elements, so I wonder if this proof can be used for **infinite purely inseparable extensions of complete fields**.