More on ONB1 identity element

“Application of Finite Fields”  page 98:

For a type I optimal normal basis, its minimal polynomial is obviously x x^n + ... + x + 1, which is irreducible over F_q if and only if n + 1 is a prime and q is primitive in Z_{n+1}

For n = 4,
x^4+x^3+x^2+x = 1
x^5 = 1
x^6 = x
x^7 = x^2
x^8 = x^3
x^9 = x^4

Ex :
Let’s prove that 1111 is the identity element of ONB1 n = 4.

1010 x 1111 = (x^8+x^2) x (x^8+x^4+x^2+x)
= x^{16}+x^{12}+x^{10}+x^9+x^{10}+x^6+x^4+x^3
= x^{16}+x^{12}+x^9+x^6+x^4+x^3
= x + x^2 + x^4 + x + x^4 + x^3
= x^2 + x^3
= x^2 + x^8
= x^8 + x^2
= 1010

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