Cauchy-Mirimanoff and related polynomials

This thesis investigates the Cauchy-Mirimanoff polynomials En and their close relatives Rn, Sn and Tn, with an emphasis on irreducibility. The Cauchy-Mirimanoff polynomials were first identified and studied by Cauchy and Liouville in 1839 in relation to Fermat's Last Theorem, but Mirimanoff in 1903 first proposed their irreducibility over Q for n a prime number. None of the standard irreducibility criteria apply directly, for example Helou showed En is always reducible modulo any prime for all odd n>=9. Computing irreducibility is problematic as the largest coefficients grow rapidly with n. The difficulty of the problem is apparent since it remains unresolved after more than 100 years. Helou, Filaseta and Beukers in 1997, Tzermias in 2007, Irick in 2010 and Lynch in 2012 have progressed the area using a range of methods, but this thesis describes an alternative original method and it is used to generalize some of the earlier results. In essence the method uses proven properties of the polynomials to reveal an inconsistency in the 2-adic or 3-adic valuation of their coefficients, depending on the polynomial under consideration. After proving several properties of the polynomials the new method is used to prove that Rm, Sm, Tm are irreducible over Q for odd m>=3, and En, Rn, Sn are irreducible over Q, for n=(2^q)m, q=1,2,3,4,5, and m>=1 odd. And using the same approach it is proved that En, for n=(3^q)m, is irreducible over Q for q=1,2,3,4 and for any odd m>=1, not divisible by 3. It is likely the results could be extended to higher values of q. It is proved that for most odd n, assuming En is reducible over Q, En is proportional to the product of two primitive irreducible polynomials over Z, both of which share all 6 of the automorphisms of En. Mirimanoff's conjecture is also investigated. Tzermias has shown Ep is irreducible for every prime p less than 1,000 and by considering a decomposed form of Ep this result is extended here to all primes less than 10,000. Several new irreducibility criteria and Theorems relating to the number and size of factors over Q are proved including some based on the approach of Schonemann which are shown to be effective. A good approximation to all the roots of En (for all n) is proved. For prime n this is used to prove an upper bound on the number of factors of En over Q. Using the estimate it is shown that for large n the roots take the form of simple linear fractional functions of the n'th roots of unity. The Newton polygon of En is studied for composite n. It is shown that if p is any prime divisor of n, then En is always reducible over the field of p-adics Qp. Irreducible factors in Qp[x] of En, Rn, Sn, Tn, are identified for n=Kp^r, where p is any prime, with 1=1. A complete factoring of En into irreducibles in Qp[x] is given for n=2p^r. Similar results are proved for Rn, Sn and Tn