About the discriminant of pure elds

The computation of the discriminant of an algebraic number eld has always been one of the most challengingproblems in algebraic number theory. Here we deal with calculating this main invariant for pure number elds (or briefly pure fields). Landsberg [۵] was one of the rst who explicitly formulated the discriminantfor pure prime degree number elds which form one of the types of algebraic number elds. Since then, thisproblem has been an attractive research subject and many formulas has been given for di erent kinds ofalgebraic number elds (see for example [۱, ۳]). We survey some results for the discriminant of some pure elds K, including K = Q(d) of degree n where the integer d is such that for each prime p dividing neither p - d or the highest power of p dividing d is coprime to p. We see that the discriminant of such K isachieved by the prime powers dividing d, n.It is known that an algebraic number eld K is generated by a single element. If K is of degreen = [K : Q], then there exists an algebraic number θ of degree n, such that K = Q(θ). By a pure numberfield we mean an algebraic number eld of the type Q() , where Xn - d is the de ning polynomial ofθ = d (see [۶]). Assume that K = Q(θ) is a prime number eld of degree n with θ = d such that foreach prime p dividing n either p - d or the highest power of p dividing d is coprime to p. It is noted that thiscondition is satis ed when d, n are coprime or d is square-free. Gassert [۲] calculated the discriminant ofsuch K with using Montes Algorithm. In [۴], the classical Theorem of Ore [۷] is applied to obtain a formulafor dK. In this approach, two concepts of p-adic valuations and p-Newton polygons for prime number p playthe most important role. Let us briey present these basic notions.Let p be a prime number. The mapping vp : Q →! ZU{∞} is called the p-adic valuation of Q wheneverfor any any nonzero integer x, vp(x) = m if pm divides x, but pm+۱ does not divide x. We extend thisde nition to any nonzero rational number x=y, by vp(x=y) = vp(x) - vp(y), where x; y ϵ Z, y ≠ ۰. Weset vp(۰) = ∞ with the convention that x < ∞and x +∞ = ∞+ x = ∞+∞ = ∞, for every integer x.Therefore, for any non-zero integer x the p-adic value vp(x) of x is de ned to be the highest power of theprime p dividing x. A p-adic valuation induces a p-adic distance dp(x; y) = p-vp(x-y) on the eld Q. Thecompletion of Q relative to the p-adic distance is again a eld, denoted by Qp and its elements are calledp-adic numbers.With using p-adic valuations, the following theorem gives a formula for the discriminant of K = Q( n pd)by the prime powers dividing d and n [۴].

کلیدواژه ها:

Pure elds ، discriminants of algebraic number elds