Thesis Title

Integral Basis of Pure Cubic Fields

Author

Kristen Bieda

Date of Graduation

Spring 2004

Degree

Master of Science in Mathematics

Department

Mathematics

Committee Chair

Clayton Sherman

Keywords

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

Subject Categories

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)

Copyright

© Kristen Bieda

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