#### Thesis Title

Integral Basis of Pure Cubic Fields

Spring 2004

#### Degree

Master of Science in Mathematics

Mathematics

Clayton Sherman

#### Keywords

algebraic number theory, complex number, pure cubic number fields, integral basis, algebraic integer

Mathematics

#### Abstract

Algebraic number theory is essentially the study of number fields, which are subfields of the complex numbers that are finite extensions of the field of rational numbers. A pure cubic number field Q[³√m], m cubefree, consists of polynomials with rational coefficients evaluated at [³√m]. An algebraic integer is a complex number that is a root of a monic polynomial with integer coefficients, and the set of algebraic integers in Q[³√m] forms the ring of integers of Q[³√m]. We can define a basis of this ring of integers, called an integral basis of Q[³√m], and any linear combination of elements from this basis will produce an algebraic integer. For m=hk², where h and k are relatively prime and squarefree and for α = ³√m, we prove that there exists an integral basis for Q[³√m] of the form: 1, α, α²/k if m ±1 (mod 9) 1, α, α² ± k²α + k² /3k if m ≡ ±1 (mod 9)