On almost <i>p</i>-rational characters of p ? p^{\prime} -degree

Abstract Let p be a prime and let G finite group. A complex character of is called almost -rational if its values belong to cyclotomic field ? ? ( e 2 ? i / n stretchy="false">) {{\mathbb{Q}}(e^{2\pi i/n})} for some ? ? + {n\in{\mathbb{Z}}^{+}} not divisible by p {p^{2}} . We prove that, in contrast usual characters, there are “many” irreducible characters groups. obtain both explicit asymptotic bounds the number terms In fact, motivated McKay–Navarro conjecture, we same bound such ? {p^{\prime}} -degree minimal situation, -characters coincides with that ???? G P {{\mathbf{N}}_{G}(P)} Syl ? {P\in{\operatorname{Syl}}_{p}(G)} Lastly, propose new way detect cyclicity Sylow -subgroups group from table, using blockwise refinement conjecture.