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In this article, we study theK1of tiled rings over local rings. Our results generalize some existing ones obtained by Keating and Xi. [ABSTRACT FROM PUBLISHER]

Let F be a totally real number field with degree n = [F:] ≥ 3. Mazur and Urbanowicz proved that if [image omitted] and F is not [image omitted] or [image omitted], then F must be one of the 14 cases listed in Mazur and Urbanowicz (1992). In this article, it is proved that 3 of these 14 cases don't satisfy (*), while all the other cases satisfy (*). Hence we find all totally real number fields which satisfy (*). [ABSTRACT FROM AUTHOR]

In this paper, it is proved that if F is a global field, then for any integer n > 3, there is an extension field E over F of degree n such that K 2 E is not generated by the Steinberg symbols { a , b } with a ? F *, b , ? E *. If however, F is a number field and D is a finite-dimensional central division F -algebra with square free index, then K 2 D is always generated by the Steinberg symbols { a , b } with a ? F *, b ? D *. Finally, the tame kernels of central division algebras over F are expressed explicitly. [ABSTRACT FROM AUTHOR]

In this paper, it is proved that the tame kernel of [Qopf](ζ8) is trivial. [ABSTRACT FROM AUTHOR]

In this article, we prove that an imaginary quadratic field F has the ideal class group isomorphic to /2 ⊕ /2 if and only if the Ono number of F is 3 and F has exactly 3 ramified primes under the Extended Riemann Hypothesis (ERH). In addition, we give the list of all imaginary quadratic fields with Ono number 3. [ABSTRACT FROM AUTHOR]