# Finite dimensional division algebra over pseudo-algebraic closed field

Is it true that any finite dimensional division algebra over a pseudo-algebraically closed field is trivial? We know that this is true for algebraically closed field.

• Presumably the answer below has to do with the following issue. Some mathematicians use "division algebra over $k$" to mean a finite dimensional $k$-algebra that is a division algebra and whose center equals $k$. That is presumably also what you intend. However, other mathematicians make no such hypothesis about the center of the division algebra. In that case, also field extensions of $k$ are "division algebras over $k$". The term "central simple algebras" clarifies this, because "central" is part of the name. Commented Aug 24, 2017 at 13:03
• @JasonStarr Yeah, if I remember correctly, EGA or SGA makes this distinction between R-rings, R-algebras, and R-extensions, the first being noncentral ring maps, the second being central ring maps, and the third being central maps into commutative rings in the section about something involving derived functors, possibly the degree 1 Quillen-Andre ones. Commented Aug 24, 2017 at 16:39
• Oops ran out of time to edit it, but yeah it's in the section on ExAlComm Commented Aug 24, 2017 at 16:45
• $\operatorname{EGA}_0 \S 18$ got it for ya Commented Aug 24, 2017 at 17:11

• It seems from the proof of Proposition $6.2.3$, page $171$ of "Central simple algebras and Galois cohomology" by Gille and Szamuely, that this is true for $C_1$-fields. But I am not an expert. Commented Aug 24, 2017 at 12:53
• No. As mentioned in the answer, finite fields are $C_1$, and every extension field of them is such an algebra with itself (so not the base field) as center.