A construction of infinite set of rational self-equivalences

In [5] it was shown that two number fields have isomorphic Witt rings of quadratic forms if and only if there is a Hilbert symbol equivalence between them. A Hilbert symbol equivalence between two number fields K and L is a pair of maps(t,T), where t: K ^∗/K^∗2→L^∗/L^∗2 is a group isomorpism and T: Ω_K→Ω _L is a bijection between the sets of finite and infinite primes of K and L, respectively, such that the Hilbert symbols are preserved: for any a; b∈K^∗=K^∗2and for any P∈Ω_K,(a; b)_P= (t(a), t(b))_T(P) A Hilbert symbol equivalence between the field Q and itself is called rational self-equivalence. In [5] the authors present a construction of equivalence of two fields starting from the so called Hilbert small equivalence of two fields. We use this idea for constructing infinite set of rational self-equivalences.