(Lagrange interpolation formula.) For n ≥ 1, let α 1 ,..., α n be n distinct elements of F q , and let β 1 ,..., β n be n arbitrary elements of F q . Show that there exists exactly one polynomial f (...

1. 
   

   
   (Lagrange interpolation formula.) For
   
   n
   
   ≥ 1, let
   
   α
   
   
   ₁
   
   
   ,...,
   
   
   α
   
   
   
   _n
   
   
   be
   
   n
   
   
   
   distinct
   

   
   

elements of

F



_q


, and let

β


₁


,...,


β



_n


be

n

arbitrary elements of

F



_q


. Show that there exists exactly one polynomial

f



(

x



) ∈

F



_q






[

x



] of degree ≤

n



− 1 such that

f

(

α



_i


) =

β



_i


for

i

= 1

,...,


n

. Furthermore, show that this polynomial is given by

n



<sub>β




i</sub>



n

f



(

x



) =

⁾

(

x



−

α



_k






)

,

n

<sup>i



=


1



g</sup>


t

<sup>(



α




i



)</sup>


k



1

=

k

/=

i

where

g


^t

(

x



) denotes the derivative of

g

(

x



) :=

<sup>n



n</sup>

k

=1