Large algebraic integers

Algebraic Numbers and Algebraic Integers

c © W W L Chen, 1984, 2013. This chapter originates from material used by the author at Imperial College London between 1981 and 1990. It is available free to all individuals, on the understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied, with or without permission from the author. However, this document may not be kept on any information storage an...

Primitive Recursive Decidability for Large Rings of Algebraic Integers

Algebraic integers of small discriminant

(1) D(α) = D(K)(I(α))2, where I(α) = [OK : Z[α]], the index of the ring generated by α in the full ring of integers OK of K. Classically, the index of the field K is the greatest common divisor of all indices I(α) for α ∈ OK with K = Q(α). Dedekind was the first to show that the index may not be 1 by exhibiting certain cubic and quartic fields with this property. By results of Bauer and von Żyl...

Chromatic roots as algebraic integers

A chromatic root is a zero of the chromatic polynomial of a graph. At a Newton Institute workshop on Combinatorics and Statistical Mechanics in 2008, two conjectures were proposed on the subject of which algebraic integers can be chromatic roots, known as the “α + n conjecture” and the “nα conjecture”. These say, respectively, that given any algebraic integer α there is a natural number n such ...

Arithmetic of Large Integers

Out[27]= 5 732 633 648 444 029 959 171 112 534 135 601 537 904 527 274 424 900 382 532 001 737 941 474 619 400 523 520 462 230 330 041Ö 125 536 677 017 734 855 696 154 228 338 013 057 234 856 311 747 239 817 067 594 543 153 099 451 342 467 908 368 438 222 191Ö 627 068 172 197 302 715 770 028 566 855 392 342 976 305 724 205 594 772 269 515 365 592 152 841 739 667 322 895 223 024 545Ö 931 314 017...