  The resolvent degree $\textrm{rd}_{\mathbb{C}}(n)$ is the smallest integer
$d$ such that a root of the general polynomial $$f(x) = x^n + a_1 x^{n-1} +
\ldots + a_n$$ can be expressed as a composition of algebraic functions in at
most $d$ variables with complex coefficients. It is known that
$\textrm{rd}_{\mathbb{C}}(n) = 1$ when $n \leqslant 5$. Hilbert was
particularly interested in the next three cases: he asked if
$\textrm{rd}_{\mathbb{C}}(6) = 2$ (Hilbert's Sextic Conjecture),
$\textrm{rd}_{\mathbb{C}}(7) = 3$ (Hilbert's 13th Problem) and
$\textrm{rd}_{\mathbb{C}}(8) = 4$ (Hilbert's Octic Conjecture). These problems
remain open. It is known that $\textrm{rd}_{\mathbb{C}}(6) \leqslant 2$,
$\textrm{rd}_{\mathbb{C}}(7) \leqslant 3$ and $\textrm{rd}_{\mathbb{C}}(8)
\leqslant 4$. It is not known whether or not $\textrm{rd}_{\mathbb{C}}(n)$ can
be $> 1$ for any $n \geqslant 6$.
  In this paper, we show that all three of Hilbert's conjectures can fail if we
replace $\mathbb C$ with a base field of positive characteristic.
