= Solution
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 $u_1,\ldots,u_m$ of $L$. Given a homogeneous $f\in L[X_1,\ldots,X_N]$ of degree $d$ with $N>d$, substitute $X_i=\sum_jx_{ij}u_j$ and take the <field norm>:
$$
F((x_{ij}))=N_{L/K}\left(f\left(\sum_jx_{1j}u_j,\ldots,\sum_jx_{Nj}u_j\right)\right).
$$
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 $f(X_1,\ldots,X_N)=0$. \b[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.
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