Attaining the exponent $5\mathrm{/}4$ for the sum-product problem in finite fields

We improve the exponent in the finite field sum-product problem from $11\mathrm{/}9$ to $5\mathrm{/}4$, improving the results of Rudnev, Shakan and Shkredov. That is, we show that if $A\subset {\mathbb{F}}_p$ has cardinality $\mathrm{\mid }A\mathrm{\mid }\ll p^{1\mathrm{/}2}$ then $$\max \{\mathrm{\mid }A\pm A\mathrm{\mid },\mathrm{\mid }AA\mathrm{\mid }\}\gtrsim \mathrm{\mid }A{\mathrm{\mid }}^{\frac{5}{4}}$$ and $$\max \{\mathrm{\mid }A\pm A\mathrm{\mid },\mathrm{\mid }A\mathrm{/}A\mathrm{\mid }\}\gtrsim \mathrm{\mid }A{\mathrm{\mid }}^{\frac{5}{4}}\mathrm{.}$$