Question

Question: Show that X³ -3 is irreducible in Z₇[X].

Short answer

Answer:

It is proved that X³ -3 is irreducible.

Step-by-step solution

Determine X3 -3

Consider the given function , X³ -3 in Z₇[X] , this is done by contradiction

Now, using Theorem 4.11, X³ -3 is written as a product of two polynomials of degree less than . Thus, the possibilities are:

Here, a,b,c,d,e, f are real numbers because,

Result

Here, X³ -3 has no roots in Z₇and thus, factorization does not contain the linear factors contradiction.

Therefore, X³ -3 is irreducible.

Hence, it is proved.