Hi guys, im stuck on a question which is:
Prove that f(x)= x^4 + x + 2 is irreducible over Z3
Note: I show that the linear factors for f(a) over Z3 are not equal to zero,
But how would i determine quadratic factors?(help solve) thanks.
Let and .
Then it turns out that for each .
However, the two polynomials are not the same!
What is a polynomial? Here is one formal definition. Let be a field. Define - an infinite tuple of coordinates. A polynomial is such that (at the -th coordinate) satisfies for all but finitely many . So when we write we mean and when we write we mean and so on. We say two polynomial are equal if and only if i.e. if and only if the two coordinates match.
I think your confusion, dear, is that you are confusing polynomial for polynomial function. A polynomial is an abstract meaning as defined above. A polynomial function is a function consisting of powers of x. Now if two polynomial functions agree at all points then the two polynomial functions are equal. However, if two polynomials (abstract) agree at all the points does not mean they are the same.
I give you an example. The polynomial, (known as Artin-Schreir polynomial - kinda famous) over can be shown to be irreducible, however, for all . This does not mean that .