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Omni-MATH / Let p>2 be a prime number. F_(p)[x] is defined as the set of all polynomials in x with coefficients in F_(p) (the integers modulo p with usual addition and subtraction), so that tw…

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Let p>2p>2 be a prime number. Fp[x]\mathbb{F}_{p}[x] is defined as the set of all polynomials in xx with coefficients in Fp\mathbb{F}_{p} (the integers modulo pp with usual addition and subtraction), so that two polynomials are equal if and only if the coefficients of xkx^{k} are equal in Fp\mathbb{F}_{p} for each nonnegative integer kk. For example, (x+2)(2x+3)=2x2+2x+1(x+2)(2 x+3)=2 x^{2}+2 x+1 in F5[x]\mathbb{F}_{5}[x] because the corresponding coefficients are equal modulo 5 . Let f,g∈Fp[x]f, g \in \mathbb{F}_{p}[x]. The pair (f,g)(f, g) is called compositional if f(g(x))≡xp2−xf(g(x)) \equiv x^{p^{2}}-x in Fp[x]\mathbb{F}_{p}[x]. Find, with proof, the number of compositional pairs (in terms of pp ).
Plain-text mathematical notation (without MathML)
Let p>2 be a prime number. F_(p)[x] is defined as the set of all polynomials in x with coefficients in F_(p) (the integers modulo p with usual addition and subtraction), so that two polynomials are equal if and only if the coefficients of x^(k) are equal in F_(p) for each nonnegative integer k. For example, (x+2)(2x+3)=2x²+2x+1 in F₅[x] because the corresponding coefficients are equal modulo 5 . Let f,g∈F_(p)[x]. The pair (f,g) is called compositional if f(g(x))≡x^(p²)−x in F_(p)[x]. Find, with proof, the number of compositional pairs (in terms of p ).
Original LaTeX notation
Let $p>2$ be a prime number. $\mathbb{F}_{p}[x]$ is defined as the set of all polynomials in $x$ with coefficients in $\mathbb{F}_{p}$ (the integers modulo $p$ with usual addition and subtraction), so that two polynomials are equal if and only if the coefficients of $x^{k}$ are equal in $\mathbb{F}_{p}$ for each nonnegative integer $k$. For example, $(x+2)(2 x+3)=2 x^{2}+2 x+1$ in $\mathbb{F}_{5}[x]$ because the corresponding coefficients are equal modulo 5 . Let $f, g \in \mathbb{F}_{p}[x]$. The pair $(f, g)$ is called compositional if $$f(g(x)) \equiv x^{p^{2}}-x$$ in $\mathbb{F}_{p}[x]$. Find, with proof, the number of compositional pairs (in terms of $p$ ).

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task_id
62496abf-de6f-5395-b174-5356aeb47dab
task_key
test--62496abf-de6f-5395-b174-5356aeb47dab
task_revision_id
1
config / split
/ test

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