Why cannot a reducible polynomial (one that has factors besides 1 and itself) be used as the polynomial to generate a finite field?
That is to say, why do we always need an irreducible polynomial as a necessary condition to generate a finite field?
In other words, how do I prove that I cannot have a finite field if the polynomial used to construct the field is not an irreducible one?
Any detailed and logical explanation would be of much help.
A field is in particular an integral domain. The ideal generated by a reducible polynomial is not a prime ideal, and therefore the quotient ring is not integral and thus cannot be a field.