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Let $p \equiv 1 \bmod 3$ be a prime, $\mathbb{F}_p$ be the finite field with $p$ elements, and $a_0$ be a generator of $\mathbb{F}_p^{\times}$ $(\mathbb{F}_p^{\times}$ the group of nonzero elements of $\mathbb{F}_p$ under multiplication.) Suppose that $f(x) = x^3 + a_2 x^2 + a_1 x + a_0$ is irreducible in $\mathbb{F}_p[x]$. Let $C = \{ c \in \mathbb{F}_p\,|\, c = b^3, \, \text{with}\, b \in \mathbb{Z}_p \}.$ For $i = 0, 1, 2$, let $D_i = \{ x \in \mathbb{F}_p \,|\, f(x)\in a_0^iC\}$.

How can we determine the cardinality of $D_i$?

Let $p \equiv 1 \bmod 3$ be a prime, $\mathbb{F}_p$ be the finite field with $p$ elements, and $a_0$ be a generator of $\mathbb{F}_p^{\times}$ $(\mathbb{F}_p^{\times}$ the group of nonzero elements of $\mathbb{F}_p$ under multiplication.) Suppose that $f(x) = x^3 + a_2 x^2 + a_1 x + a_0$ is irreducible in $\mathbb{F}_p[x]$. Let $C = \{ c \in \mathbb{F}_p\,|\, c = b^3, \, \text{with}\, b \in \mathbb{F}_p \}.$ For $i = 0, 1, 2$, let $D_i = \{ x \in \mathbb{F}_p \,|\, f(x)\in a_0^iC\}$.

How can we determine the cardinality of $D_i$?

Let $p \equiv 1 \bmod 3$ be a prime, $\mathbb{F}_p$ be the finite field with $p$ elements, and $a_0$ be a generator of $\mathbb{F}_p^{\times}$ $(\mathbb{F}_p^{\times}$ the group of nonzero elements of $\mathbb{F}_p$ under multiplication. Suppose that $f(x) = x^3 + a_2 x^2 + a_1 x + a_0$ is irreducible in $\mathbb{F}_p[x]$. Let $C = \{ c \in \mathbb{F}_p\,|\, c = b^3, \, \text{with}\, b \in \mathbb{Z}_p \}.$ For $i = 0, 1, 2$, let $D_i = \{ x \in \mathbb{F}_p \,|\, f(x)\in a_0^iC\}$.

How can we determine the cardinality of $D_i$?

Let $p \equiv 1 \bmod 3$ be a prime, $\mathbb{F}_p$ be the finite field with $p$ elements, and $a_0$ be a generator of $\mathbb{F}_p^{\times}$ $(\mathbb{F}_p^{\times}$ the group of nonzero elements of $\mathbb{F}_p$ under multiplication.) Suppose that $f(x) = x^3 + a_2 x^2 + a_1 x + a_0$ is irreducible in $\mathbb{F}_p[x]$. Let $C = \{ c \in \mathbb{F}_p\,|\, c = b^3, \, \text{with}\, b \in \mathbb{Z}_p \}.$ For $i = 0, 1, 2$, let $D_i = \{ x \in \mathbb{F}_p \,|\, f(x)\in a_0^iC\}$.

How can we determine the cardinality of $D_i$?

Let $p \equiv 1 \bmod 3$ be a prime and $a_0$ be a generator of $\mathbb{F}_p^{\times}$. Suppose that $f(x) = x^3 + a_2 x^2 + a_1 x + a_0$ is irreducible in $\mathbb{F}_p[x]$. Let $C = \{ c \in \mathbb{F}_p\,|\, c = b^3, \, \text{with}\, b \in \mathbb{Z}_p \}.$ For $i = 0, 1, 2$, let $D_i = \{ x \in \mathbb{F}_p \,|\, f(x)\in a_0^iC\}$.

How can we determine the cardinality of $D_i$?

Let $p \equiv 1 \bmod 3$ be a prime, $\mathbb{F}_p$ be the finite field with $p$ elements, and $a_0$ be a generator of $\mathbb{F}_p^{\times}$ $(\mathbb{F}_p^{\times}$ the group of nonzero elements of $\mathbb{F}_p$ under multiplication. Suppose that $f(x) = x^3 + a_2 x^2 + a_1 x + a_0$ is irreducible in $\mathbb{F}_p[x]$. Let $C = \{ c \in \mathbb{F}_p\,|\, c = b^3, \, \text{with}\, b \in \mathbb{Z}_p \}.$ For $i = 0, 1, 2$, let $D_i = \{ x \in \mathbb{F}_p \,|\, f(x)\in a_0^iC\}$.

How can we determine the cardinality of $D_i$?