ABSTRACT
Let R be a Noetherian integral domain, R[X ] be a polynomial ring, and K be the quotient field of R. Let α be an element of an algebraic field extension L of K and π : R[X ] −→ R[α] be the R-algebra homomorphism sending X to α. Let ϕα(X ) be the monic minimal polynomial of α over K with deg ϕα(X ) = d and write ϕα(X ) = Xd + η1 Xd−1 + · · · + ηd . Let I[α] :=
⋂d i=1(R :R ηi )
(= R[X ] :R ϕ(X )). For f (X ) ∈ R[X ], let c( f (X )) denote the ideal generated by the coefficients of f (X ). Let J[α] := I[α]c(ϕα(X )), which is an ideal of R and contains I[α]. The element α is called an anti-integral element of degree d over R if Ker(π ) = I[α]ϕα(X )R[X ]. When α is an anti-integral element over R, R[α] is called an anti-integral extension of R. In the case K (α) = K , an anti-integral element α is the same as an anti-integral element (i.e., R = R〈α〉) defined in Chapter 1. The element α is called a super-primitive element of degree d over R if J[α] ⊂ p for all primes p of depth 1.
