Consider the polynomial ring F [ x ] over a field F. Let d and...

Luz Stokes

Luz Stokes

Answered

2022-07-19

Consider the polynomial ring F [ x ] over a field F. Let d and n be two nonnegative integers.
Prove: x d 1 x n 1

Answer & Explanation

Sandra Randall

Sandra Randall

Expert

2022-07-20Added 17 answers

Suppose x d 1 x n 1. By division algorithm, we can write n = q d + r for some q , r N 0 with 0 r < d. Now, observe that
x d 1 ( x d 1 ) ( x n d + x n 2 d + + x n q d + 1 )
Expanding the above, and cancelling many terms, we get that
x d 1 x n + x d x n q d 1 = x n 1 + x d x r
Together with x d 1 x n 1, this implies:
x d 1 x d x r = ( x d 1 ) + ( 1 x r )
which gives x d 1 x r 1. This is a contradiction, unless r = 0, in which case d n
The converse easily follows from the identity x n 1 = ( x 1 ) ( x n 1 + x n 2 + + x + 1 )

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