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A $C_1$ field is a Ci field with the bound $N>d$. The Chevalley-Warning theorem gives finite-field examples; the Tsen theorem gives function fields of curves over algebraically closed fields. A $C_1$ field has trivial Brauer group.

For the function field $K$ of a curve over a finite full constant field $k$, extend constants to $\overline{k}$. The resulting curve function field is $C_1$ by the Tsen theorem, so every central simple algebra splits there. The finite collection of coefficients of a splitting isomorphism descends to some finite constant extension $k^{\prime }/k$. Regularity makes $K{k}^{\prime }/K$ cyclic, giving a finite cyclic splitting field.

For a nonnegative integer $r$, a Cr field is a field over which every homogeneous polynomial of positive degree $d$ in $N>d^r$ variables has a nontrivial zero. Thus a C1 field uses the bound $N>d$, and a C2 field uses $N>d^2$.

The required field theorems are as follows. The Chevalley-Warning theorem implies that every finite field is $C_1$: the number of zeros of a polynomial with degree smaller than its number of variables is divisible by the characteristic, and for a homogeneous polynomial the origin is already a zero. The Lang-Nagata theorem for Ci fields states that a finitely generated extension of transcendence degree $s$ of a $C_r$ field is $C_{r+s}$. The Tsen theorem states that the function field of a curve over an algebraically closed field is $C_1$. In particular, finite fields, algebraically closed fields and fields such as $\mathbb{C}(t)$ are examples of $C_1$ fields. A function field of one variable over a finite field is $C_2$ by the Lang theorem. These are statements of the theorems; no theorem proof is needed here.

For the requested finite extension stability of C1 fields, let $m=[L:K]$ and choose a $K$-basis $u_1,\dots ,u_m$ of $L$. Given a homogeneous $f\in L[X_1,\dots ,X_N]$ of degree $d$ with $N>d$, substitute $X_i={\sum }_j{x}_{ij}{u}_j$ and take the field norm:The field norm is the determinant of multiplication on the $m$-dimensional $K$-space $L$, hence a homogeneous polynomial of degree $m$ in its coordinates. Consequently $F$ is homogeneous of degree $md$ in $mN>md$ variables over $K$. The $C_1$ property gives a nonzero coordinate vector $(x_{ij})$ with $F=0$. Its corresponding vector $(X_i)\in L^N$ is nonzero, since the $u_j$ are a basis. The norm of a field element vanishes only for the zero element, so $f(X_1,\dots ,X_N)=0$. Every finite extension of a $C_1$ field is $C_1$. This proof includes inseparable finite extensions, because the determinant definition of the field norm requires no separability.

Let $k$ be the full constant field of $K$. It is a finite extension of the given finite field, hence is itself finite. Since $k$ is perfect and algebraically closed in $K$, the one-variable extension $K/k$ is regular. In a common algebraic closure, put $E=K\overline{k}$. Then $E$ is the function field of a geometrically integral curve over the algebraically closed field $\overline{k}$.

The Tsen theorem makes $E$ a $C_1$ field, and the preceding argument gives $\mathrm{Br}(E)=0$. Thus $A{\otimes }_{K}E$ is a matrix algebra. A splitting isomorphism and its inverse involve only finitely many coefficients in $E$. All these coefficients lie in $K{k}^{\prime }$ for some finite extension $k^{\prime }/k$, since $E$ is the union of these finite constant extensions. The two isomorphism identities consequently already hold over $K{k}^{\prime }$, so $A$ splits there.

Finite extensions of a finite field are cyclic, and regularity gives $K\cap k^{\prime }=k$ and preserves their degree under this scalar extension. Thereforeis cyclic. A finite cyclic extension of constants splits $A$. This is cyclic splitting by extension of constants. The argument does not assert that the splitting extension has degree exactly the index of $A$; a larger cyclic constant extension is sufficient for the requested conclusion.