Newtonian and Schinzel quadratic fields

We consider a quadratic extension of a global field and give the maximal length of a Newton sequence, that is, a simultaneous ordering in Bhargava’s sense or a Schinzel sequence, that satisfies the condition of the Brownin–Schinzel problem. In the case of a number field Q ( d ) , we show that the maximal length of a Schinzel sequence is 1, except in seven particular cases, and explicitly compute the maximal length of a Schinzel sequence in these special cases. We show that Newton sequences are also finite, except for at most finitely many cases, all real, and such that d ≡ 1 ( mod 8 ) . For d ≢ 1 ( mod 8 ) , we show that the maximal length of a Newton sequence is 1, except in five particular cases, and again explicitly compute the maximal length in these special cases. In the case of a quadratic extension of a function field F q ( T ) , we similarly show that, unless the ring of integers is isomorphic to some function field (in which case there are obviously infinite Newton and Schinzel sequences), the maximal length of a Schinzel sequence is finite and in fact, equal to q . For imaginary extensions, Newton sequences are known to be finite (unless the ring of integers is isomorphic to some function field) and we show here that the same holds in the real case, but for finitely many extensions.