Joe Root his first golden duck in his ODI career. The 5th ODI was better for Root as he scored a quick 21 before being bowled and also took the wicket of George Bailey. England lost the series 2–1 and Root finished the series with 36 runs and 1 wicket. Root was selected for the 2013-14 Ashes series. He was selected at number 6 for the first Test as Michael Carberry was given the opening batsman role alongside Alastair Cook. Root only managed 2 runs from 7 balls in the first innings before being caught off a Mitchell Johnson delivery. In the
Root of unity theorem. Every th root of unity is a primitive th root of unity for exactly one positive divisor of . This implies that This formula represents the factorization of the polynomial into irreducible factors. Applying Möbius inversion to the formula gives where is the Möbius function. So the first few cyclotomic polynomials are If is a prime number, then all the th roots of unity except 1 are primitive th roots, and we have Substituting any positive integer ≥ 2 for , this sum becomes a base repunit. Thus a necessary (but not sufficient) condition for a repunit to be