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We study the irreducible quotient $\mathcal{L}_{t,c}$ of the polynomialrepresentation of the rational Cherednik algebra$\mathcal{H}_{t,c}(S_n,\mathfrak{h})$ of type $A_{n-1}$ over an algebraicallyclosed field of positive characteristic $p$ where $p|n-1$. In the $t=0$ case,for all $c\ne 0$ we give a complete description of the polynomials in themaximal proper graded submodule $\ker \mathcal{B}$, the kernel of thecontravariant form $\mathcal{B}$, and subsequently find the Hilbert series ofthe irreducible quotient $\mathcal{L}_{0,c}$. In the $t=1$ case, we give acomplete description of the polynomials in $\ker \mathcal{B}$ when thecharacteristic $p=2$ and $c$ is transcendental over $\mathbb{F}_2$, and computethe Hilbert series of the irreducible quotient $\mathcal{L}_{1,c}$. In doingso, we prove a conjecture due to Etingof and Rains completely for $p=2$, andalso for any $t=0$ and $n\equiv 1\pmod{p}$. Furthermore, for $t=1$, we prove asimple criterion to determine whether a given polynomial $f$ lies in $\ker\mathcal{B}$ for all $n=kp+r$ with $r$ and $p$ fixed.

The Hilbert Series of the Irreducible Quotient of the Polynomial  Representation of the Rational Cherednik Algebra of Type $A_{n-1}$ in  Characteristic $p$ for $p|n-1$