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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.

Every sequence of $4p-3$ elements of ${\mathbb{F}}_p^2$, where $p$ is prime, contains $p$ terms whose sum is zero. A proof applies the Chevalley-Warning theorem to the cardinality and two coordinate sums and then double-counts zero-sum subsequences.

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.

For a sequence $J$ in ${\mathbb{F}}_p^2$, write $(r\mid J)$ for the number of $r$-term subsequences whose sum is zero. We use the following consequence of the Chevalley-Warning theorem: if $|J|=3p-1$, thenIndeed, with one variable $x_i$ for each term $(a_i,b_i)$, apply Chevalley--Warning toTheir degree sum is $3(p-1)<3p-1$. A common zero has support of size $0,p$, or $2p$ modulo $p$, and each fixed support contributes $(p-1{)}^{|S|}$ assignments. Reducing modulo $p$ gives the displayed congruence.

Now let the given $3p$ terms have total sum zero. If no $p$ of them summed to zero, delete any one term and apply the congruence to the remaining $3p-1$ terms. It givesso those remaining terms contain a zero-sum $2p$-subsequence. Its complement in the original $3p$ terms has size $p$ and sum zero, contradicting the assumption. Therefore the required $p$ terms exist.