Hello Codeforcers! As a student of department of mathematics, I'd like to demonstrate an application of Extended Field.
First we should include the idea of splitting field, a splitting field F of a polynomial P is the smallest field such that
I use the word "smallest" with the sense that if P split over E, .
In the other word, P decomposes into linear factor under it's splitting field.
For example:
P = x2 + 1 does not split over but split over where which is widely used as complex.
Let p = 998244353 = 7 * 17 * 223 + 1, P = x224 - 1 does not split over but split over where . The multiplication order ord(w2) = 223.
Let p = 109 + 7, Fibonacci number can be easily written as . Where φ, ψ are both root of (Remark that x2 = 5 has no root in Z_p)
The problem to be demonstrated is 718C - Саша и массив, which correspond to the third example.
In the problem, we should handle Fibonacci number 1, 1, 2, 3, 5, 8, ... with the formula
Once a splitting field implementation is done, we can FFT under Zp more than. I haven't implemented it yet.