The minimum discriminant of totally real algebraic number fields of degree 9 with cubic subfields

Abstract

With the help of the computer language UBASIC86, the minimum discriminant d ( K ) d(K) of totally real algebraic number fields K of degree 9 with cubic subfields F is determined. It is given by d ( K ) = 16240385609 d(K) = 16240385609 . The defining equation for K is given by f ( x ) = x 9 − x 8 − 9 x 7 + 4 x 6 + 26 x 5 − 2 x 4 − 25 x 3 − x 2 + 7 x + 1 f(x) = {x^9} - {x^8} - 9{x^7} + 4{x^6} + 26{x^5} - 2{x^4} - 25{x^3} - {x^2} + 7x + 1 , and K is uniquely determined by d ( K ) d(K) up to Q-isomorphism. The field K has the cubic subfield F with d ( F ) = 49 d(F) = 49 defined by the polynomial f ( x ) = x 3 + x 2 − 2 x − 1 f(x) = {x^3} + {x^2} - 2x - 1 .