Interesting. So the proof is to show that every finite Boolean ring has even cardinality. Well the additive group of the ring can be thought of as a vector space over F2, a field consisting of two elements. If we consider only a finite case, then it must be a finite space over this field. It...
Maybe if you give a different example it would be easier to see. I don't really see much of a correlation between the general case and the example you gave. Anyways, you can always fall back on the general technique for finding irreduciblity.
Let I be a proper ideal in the integral domain R...
I hope this helps.
The polynomial x4+x3+x2+x +1 is the cyclotomic polynomial \Phi5(x). I believe the general polynomial that you described is also the monic cyclotomic polynomial \Phin(x) in Z[x] with degree \varphi(n). These polynomials are irreducible. The proof is a little tedious and not...