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1. Characters of $$\pi '$$-degree and small cyclotomic fields

   https://doi.org/10.1007/s10231-020-01025-x  

   Giannelli, Eugenio; Hung, Nguyen Ngoc; Schaeffer Fry, A. A.; Vallejo, Carolina (June 2021, Annali di Matematica Pura ed Applicata (1923 -))

   null (Ed.)

   Abstract We show that every finite group of order divisible by 2 or q , where q is a prime number, admits a $$\{2, q\}'$$ { 2 , q } ′ -degree nontrivial irreducible character with values in $${\mathbb{Q}}(e^{2 \pi i /q})$$ Q ( e 2 π i / q ) . We further characterize when such character can be chosen with only rational values in solvable groups. These results follow from more general considerations on groups admitting a $$\{p, q\}'$$ { p , q } ′ -degree nontrivial irreducible character with values in $${\mathbb{Q}}(e^{2 \pi i /p})$$ Q ( e 2 π i / p ) or $${\mathbb{Q}}(e^{ 2 \pi i/q})$$ Q ( e 2 π i / q ) , for any pair of primes p and q . Along the way, we completely describe simple alternating groups admitting a $$\{p, q\}'$$ { p , q } ′ -degree nontrivial irreducible character with rational values. 

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2. The fields of values of characters of degree not divisible by p

   https://doi.org/10.1017/fmp.2021.1  

   Navarro, Gabriel; Tiep, Pham Huu (January 2021, Forum of Mathematics, Pi)

   Abstract We study the fields of values of the irreducible characters of a finite group of degree not divisible by a prime p . In the case where $p=2$ , we fully characterise these fields. In order to accomplish this, we generalise the main result of [ILNT] to higher irrationalities. We do the same for odd primes, except that in this case the analogous results hold modulo a simple-to-state conjecture on the character values of quasi-simple groups. 

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3. On 2-superirreducible polynomials over finite fields

   https://doi.org/10.1016/j.indag.2024.08.005  

   Bober, Jonathan W; Du, Lara; Fretwell, Daniel; Kopp, Gene S; Wooley, Trevor D (May 2025, Indagationes Mathematicae)

   We investigate k-superirreducible polynomials, by which we mean irreducible polynomials that remain irreducible under any polynomial substitution of positive degree at most k. Let F be a finite field of characteristic p. We show that no 2-superirreducible polynomials exist in F[t] when p=2 and that no such polynomials of odd degree exist when p is odd. We address the remaining case in which p is odd and the polynomials have even degree by giving an explicit formula for the number of monic 2-superirreducible polynomials having even degree d. This formula is analogous to that given by Gauss for the number of monic irreducible polynomials of given degree over a finite field. We discuss the associated asymptotic behaviour when either the degree of the polynomial or the size of the finite field tends to infinity. 

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4. Irreducible characters of even degree and normal Sylow 2-subgroups

   https://doi.org/10.1017/S0305004116000669  

   HUNG, NGUYEN NGOC; TIEP, PHAM HUU (March 2017, Mathematical Proceedings of the Cambridge Philosophical Society)

   Abstract The classical Itô-Michler theorem on character degrees of finite groups asserts that if the degree of every complex irreducible character of a finite group G is coprime to a given prime p , then G has a normal Sylow p -subgroup. We propose a new direction to generalize this theorem by introducing an invariant concerning character degrees. We show that if the average degree of linear and even-degree irreducible characters of G is less than 4/3 then G has a normal Sylow 2-subgroup, as well as corresponding analogues for real-valued characters and strongly real characters. These results improve on several earlier results concerning the Itô-Michler theorem. 

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5. The inductive McKay–Navarro conditions for the prime 2 and some groups of Lie type

   https://doi.org/10.1090/bproc/123  

   Ruhstorfer, L.; Schaeffer Fry, A. (April 2022, Proceedings of the American Mathematical Society, Series B)

   For a prime ℓ \ell , the McKay conjecture suggests a bijection between the set of irreducible characters of a finite group with ℓ ′ \ell ’ -degree and the corresponding set for the normalizer of a Sylow ℓ \ell -subgroup. Navarro’s refinement suggests that the values of the characters on either side of this bijection should also be related, proposing that the bijection commutes with certain Galois automorphisms. Recently, Navarro–Späth–Vallejo have reduced the McKay–Navarro conjecture to certain “inductive” conditions on finite simple groups. We prove that these inductive McKay–Navarro (also called the inductive Galois–McKay) conditions hold for the prime ℓ = 2 \ell =2 for several groups of Lie type, namely the untwisted groups without non-trivial graph automorphisms. 

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