Joe Root to an outstanding innings of 156 runs from Aaron Finch. Root came in with England 37/3 and he managed to score 90* off 49 balls but England lost by 39 runs. In the second T20I Joe batted at number 7 but only faced 1 ball scoring 1 run before England's innings finished. England won the match and drew the series, Root finished with 91 runs (both matches he was not out) and 1 wicket. The third ODI was abandoned in England's 15th over, Root had made 12 runs before being caught. In the fourth ODI Root was bowled for 0,
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