Integral Basis of Pure Cubic Fields
Date of Graduation
Master of Science in Mathematics
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)
algebraic number theory, complex number, pure cubic number fields, integral basis, algebraic integer
© Kristen Bieda
Bieda, Kristen, "Integral Basis of Pure Cubic Fields" (2004). MSU Graduate Theses. 1612.