Consider polynomials in one variable over the finite field F_{2} with 2 elements. Show that if n + 1 is not prime, then 1 + x + x^{2} + ... + x^{n} is reducible. Can it be reducible if n + 1 is prime?

**Solution**

Let n+1 = ab, then 1 + x + x^{2} + ... + x^{n} = (1 + x + x^{2} + ... + x^{a-1})(1 + x^{a} + x^{2a} + ... + x^{ab-a}). Note that this does not depend upon the field having two elements.

Yes. For example, (1 + x + x^{3})(1 + x^{2} + x^{3}) = 1 + x + x^{2} + x^{3} + x^{4} + x^{5} + x^{6}.

© John Scholes

jscholes@kalva.demon.co.uk

15 Feb 2002