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Let K=F(x), where x is transcendental over F and F is an algebraically closed field. Does there exist a non-commutative division algebra L with the center K and [L : K] < infinity?

I think, but I'm not sure, that an old result due to Tsen implies that the answer is no? I'd like to know if there's another way, other than applying Tsen's theorem, to prove this? Thanks.

Let $K=F(x)$, where $x$ is transcendental over $F$ and $F$ is an algebraically closed field. Does there exist a non-commutative division algebra $L$ with center $K$ and $[L:K]<\infty$?

I think, but I'm not sure, that an old result due to Tsen implies that the answer is no. I'd like to know if there's another way, other than applying Tsen's theorem, to prove this. Thanks.

Let K=F(x), where x is transcendental over F and F is an algebraically closed field. Does there exist a non-commutative division algebra L with the center K and [L : K] < infinity?

I think, but I'm not sure, that an old result due to Tsen implies that the answer is no? I'd like to know if there's another way (probably more algebraic and less geometric) to prove this? Thanks.

Let K=F(x), where x is transcendental over F and F is an algebraically closed field. Does there exist a non-commutative division algebra L with the center K and [L : K] < infinity?

I think, but I'm not sure, that an old result due to Tsen implies that the answer is no? I'd like to know if there's another way, other than applying Tsen's theorem, to prove this? Thanks.

Let K=F(x), where x is transcendental over F and F is an algebraically closed field. Does there exist a finite field extension L/K such that [L : K] > 1?

I think, but I'm not sure, that an old result due to Tsen implies that the answer is no? I'd like to know if there's another way (probably more algebraic and less geometric) to prove this? Thanks.

Let K=F(x), where x is transcendental over F and F is an algebraically closed field. Does there exist a non-commutative division algebra L with the center K and [L : K] < infinity?

I think, but I'm not sure, that an old result due to Tsen implies that the answer is no? I'd like to know if there's another way (probably more algebraic and less geometric) to prove this? Thanks.