CRC polynomial must include $X^0$?

Question

Must every polynomial include $x^0$, for example $G(x) =\dots+ x^0$?

I could not find any resource which elaborates on it. From computer networks:

Both the high- and low-order bits of the generator must be 1

Answer 1

Yes, it's a basic property of the GF (generator polymonial of cyclic codes, on which CRC are based), its independent coefficient cannot be zero. That's explained in every textbook, see eg here, proof in page 3.