Fermat's Little Theorem states that if p is a prime number, then for any integer a, the number a^p - a is an integer multiple of p. In other words, a^p ≡ a (mod p).
We can use this theorem to find the zeros of the polynomial 2x²¹⁹ + 3x⁷⁴ +2x⁵⁷ +3x⁴⁴ in Z5.
First, we need to rewrite the exponents in terms of mod 4 (since 5-1=4).
For 2x²¹⁹, we have 219 ≡ 3 (mod 4), so 2x²¹⁹ ≡ 2x³ (mod 5).
For 3x⁷⁴, we have 74 ≡ 2 (mod 4), so 3x⁷⁴ ≡ 3x² (mod 5).
For 2x⁵⁷, we have 57 ≡ 1 (mod 4), so 2x⁵⁷ ≡ 2x (mod 5).
For 3x⁴⁴, we have 44 ≡ 0 (mod 4), so 3x⁴⁴ ≡ 3 (mod 5).
Now we can rewrite the polynomial as:
2x³ + 3x² + 2x + 3
To find the zeros of this polynomial in Z5, we can simply plug in each value of x from 0 to 4 and see which ones give us a result of 0.
When x=0, we get:
2(0)³ + 3(0)² + 2(0) + 3 = 3
When x=1, we get:
2(1)³ + 3(1)² + 2(1) + 3 = 10 ≡ 0 (mod 5)
So x=1 is a zero of the polynomial.
When x=2, we get:
2(2)³ + 3(2)² + 2(2) + 3 = 53 ≡ 3 (mod 5)
When x=3, we get:
2(3)³ + 3(3)² + 2(3) + 3 = 114 ≡