Normal integral bases of Lehmer's cyclic quintic fields

Let $K_n$ be a tamely ramified cyclic quintic field generated by a root of Emma Lehmer's parametric polynomial. We give all normal integral bases for $K_n$ only by the roots of the polynomial, which is a generalization of the work of Lehmer in the case that $n^4+5n^3+15n^2+25n+25$ is prime number, and Spearman-Willliams in the case that $n^4+5n^3+15n^2+25n+25$ is square-free

Chemical compositions of six metal-poor stars in the ultra-faint dwarf spheroidal galaxy Bo\"otes I

Ultra-faint dwarf galaxies recently discovered around the Milky Way (MW) contain extremely metal-poor stars, and might represent the building blocks of low-metallicity components of the MW. Among them, the Bo\"otes I dwarf spheroidal galaxy is of particular interest because of its exclusively old stellar population. We determine chemical compositions of six red giant stars in Bo\"otes I, based on the high-resolution spectra obtained with the High Dispersion Spectrograph mounted on the Subaru Telescope. Abundances of 12 elements, including C, Na, alpha, Fe-peak, and neutron capture elements, were determined for the sample stars. The abundance results were compared to those in field MW halo stars previously obtained using an abundance analysis technique similar to the present study. We confirm the low metallicity of Boo-094 ([Fe/H]=-3.4). Except for this star, the abundance ratios ([X/Fe]) of elements lighter than zinc are generally homogeneous with small scatter around the mean values in the metallicities spanned by the other five stars (-2.7-2.7 show no significant enhancement of carbon. The [Mg/Fe] and [Ca/Fe] ratios are almost constant with a modest decreasing trend with increasing [Fe/H] and are slightly lower than the field halo stars. The [Sr/Fe] and [Sr/Ba] ratios also tend to be lower in the Bo\"otes I stars than in the halo stars. Our results of small scatter in the [X/Fe] ratios for elements lighter than zinc suggest that these abundances were homogeneous among the ejecta of prior generation(s) of stars in this galaxy.Comment: 16 pages, 12 figures. Accepted to A&A, language correcte

Chebyshev's Bias against Splitting and Principal Primes in Global Fields

A reason for the emergence of Chebyshev's bias is investigated. The Deep Riemann Hypothesis (DRH) enables us to reveal that the bias is a natural phenomenon for making a well-balanced disposition of the whole sequence of primes, in the sense that the Euler product converges at the center. By means of a weighted counting function of primes, we succeed in expressing magnitudes of the deflection by a certain asymptotic formula under the assumption of DRH, which gives a new formulation of Chebyshev's bias. For any Galois extension of global fields and for any element $\sigma$ in the Galois group, we establish a criterion of the bias of primes whose Frobenius elements are equal to $\sigma$ under the assumption of DRH. As an application we obtain a bias toward non-splitting and non-principle primes in abelian extensions under DRH. In positive characteristic cases, DRH is proved, and all these results hold unconditionally