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Omni-MATH / Let p be an odd prime number, and let F_(p) denote the field of integers modulo p. Let F_(p)[x] be the ring of polynomials over F_(p), and let q(x)∈F_(p)[x] be given by …

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problem

Let pp be an odd prime number, and let Fp\mathbb{F}_p denote the field of integers modulo pp. Let Fp[x]\mathbb{F}_p[x] be the ring of polynomials over Fp\mathbb{F}_p, and let q(x)∈Fp[x]q(x) \in \mathbb{F}_p[x] be given by q(x)=∑k=1p−1akxk, q(x) = \sum_{k=1}^{p-1} a_k x^k, where \[ a_k = k^{(p-1)/2} \mod{p}. \] Find the greatest nonnegative integer nn such that (x−1)n(x-1)^n divides q(x)q(x) in Fp[x]\mathbb{F}_p[x].
Plain-text mathematical notation (without MathML)
Let p be an odd prime number, and let F_(p) denote the field of integers modulo p. Let F_(p)[x] be the ring of polynomials over F_(p), and let q(x)∈F_(p)[x] be given by q(x)=∑_(k=1)^(p−1)a_(k)x^(k), where \[ a_k = k^{(p-1)/2} \mod{p}. \] Find the greatest nonnegative integer n such that (x−1)^(n) divides q(x) in F_(p)[x].
Original LaTeX notation
Let $p$ be an odd prime number, and let $\mathbb{F}_p$ denote the field of integers modulo $p$. Let $\mathbb{F}_p[x]$ be the ring of polynomials over $\mathbb{F}_p$, and let $q(x) \in \mathbb{F}_p[x]$ be given by \[ q(x) = \sum_{k=1}^{p-1} a_k x^k, \] where \[ a_k = k^{(p-1)/2} \mod{p}. \] Find the greatest nonnegative integer $n$ such that $(x-1)^n$ divides $q(x)$ in $\mathbb{F}_p[x]$.

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Source and history

task_id
77fafa2c-0c53-5f5b-95ec-4cf17b60d04d
task_key
test--77fafa2c-0c53-5f5b-95ec-4cf17b60d04d
task_revision_id
1
config / split
/ test

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