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Romania NMO 2023 Grade 12 P2

Source: Romania National Olympiad 2023

April 14, 2023

polynomialfieldfinite fieldssuperior algebra

Problem Statement

Let

p

be a prime number,

n

a natural number which is not divisible by

p

, and

\mathbb{K}

is a finite field, with

char(K) = p, |K| = p^n, 1_{\mathbb{K}}

unity element and

\widehat{0} = 0_{\mathbb{K}}.

For every

m \in \mathbb{N}^{*}

we note

\widehat{m} = \underbrace{1_{\mathbb{K}} + 1_{\mathbb{K}} + \ldots + 1_{\mathbb{K}}}_{m \text{ times}}

and define the polynomial

f_m = \sum_{k = 0}^{m} (-1)^{m - k} \widehat{\binom{m}{k}} X^{p^k} \in \mathbb{K}[X].

a) Show that roots of

f_1

are

\left\{ \widehat{k} | k \in \{0,1,2, \ldots , p - 1 \} \right\}

.
b) Let

m \in \mathbb{N}^{*}.

Determine the set of roots from

\mathbb{K}

of polynomial

f_{m}.

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