Primes is in P, proof of hendrik Lenstra Jr. lemma

In the paper describing AKS primality test : http://annals.math.princeton.edu/wp-content/uploads/annals-v160-n2-p12.pdf

On page no. 8 Lemma 4.7 last paragraph, I cannot understand how number of distinct polynomials with degree less than $t$ is calculated.

I understood that number of distinct polynomials with degree equal to 1 is $l + 1$, but how is the formula for a general degree ($t$) is derived? Can anyone please help me with this.

Although this paper also describes an induction method to prove this on page no. 8, but I could not understand how the number of polynomials with degree $(t - 1)$ is calculated here.Link: http://www.andrew.cmu.edu/user/jpaulson/AKS.pdf

• Please post the relevant mathematical details here, and leave the link as reference for extended information. – barak manos Jun 19 '16 at 14:59
• I have written my doubt clearly. Can you please say what mathematical details are you asking for. – Mayank Jun 19 '16 at 15:01
• Any detail which is available at the given link but not within this post, AND - which is required (IYO) for answering this question. – barak manos Jun 19 '16 at 15:09
• this is better written: dms.umontreal.ca/~andrew/PDF/Bulletin04.pdf – Will Jagy Jun 19 '16 at 17:42