The resolvent degree \operatorname{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}+ \dots + a_{n} can be expressed as a composition of algebraic functions in at most d variables with complex coefficients. It is known that \operatorname{rd}_{\mathbb{C}}(n)=1 when n\leqslant 5 . Hilbert was particularly interested in the next three cases: he asked if \operatorname{rd}_{\mathbb{C}}(6)=2 (Hilbert’s Sextic conjecture), \operatorname{rd}_{\mathbb{C}}(7)=3 (Hilbert’s 13th problem) and \operatorname{rd}_{\mathbb{C}}(8)=4 (Hilbert’s Octic conjecture). These problems remain open. It is known that \operatorname{rd}_{\mathbb{C}}(6)\leqslant 2 , \operatorname{rd}_{\mathbb{C}}(7)\leqslant 3 and \operatorname{rd}_{\mathbb{C}}(8)\leqslant 4 . It is not known whether or not \operatorname{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.