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We show that every finite group of order divisible by 2 or q, where q is a prime number, admits a {2,q}′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{2, q\}'$$\end{document}-degree nontrivial irreducible character with values in Q(e2πi/q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{Q}}(e^{2 \pi i /q})$$\end{document}. 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}′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{p, q\}'$$\end{document}-degree nontrivial irreducible character with values in Q(e2πi/p)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{Q}}(e^{2 \pi i /p})$$\end{document} or Q(e2πi/q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{Q}}(e^{ 2 \pi i/q})$$\end{document}, for any pair of primes p and q. Along the way, we completely describe simple alternating groups admitting a {p,q}′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{p, q\}'$$\end{document}-degree nontrivial irreducible character with rational values.


Introduction
One of the main problems in finite group representation theory is to understand fields of values of characters, by which we mean the smallest field containing all values of a given character. A classical result of Burnside states that groups of odd order do not possess nontrivial irreducible characters with real fields of values. Actually, this property characterizes odd-order groups in an elementary way, see Theorem 2.3. It is also true that a group G has even order if, and only if, G possesses a nontrivial irreducible character with rational field of values [24,Theorem 8.2]. However, unlike the real case, the proof of this simply stated result already requires the classification of the finite simple groups [9] (CFSG for short), evidencing the deep nature of rationality phenomena in character theory.
R. Gow conjectured that every finite group of even order has a nontrivial irreducible character with odd degree and rational field of values. In 2008, G. Navarro and P.H. Tiep [24,Theorem B] finally confirmed this prediction. Later (but appearing first in the literature [23]), they generalized their result by proving that every finite group of order divisible by a prime q admits a nontrivial irreducible character of degree coprime to q with values in the rather small cyclotomic extension ℚ(e 2i ∕q ) . The study of fields of values of irreducible characters of degree not divisible by a given prime is a subject interesting in its own right [15], which is recently blooming thanks to a growing interest in the Galois refinement of the McKay conjecture proposed by G. Navarro in [20]. This refined conjecture has been reduced to a question on simple groups in [21], which makes it important to understand the fields of values for such groups.
Usually, extensions of results from one prime to a set of primes fail (at least without assuming separability properties in the group), and the behavior of finite groups with respect to properties related to is no longer smooth. However, in [8] the first-, third-, and fourth-named authors show that every nontrivial group possesses a nontrivial irreducible character of degree not divisible by any prime in , where is any set consisting of at most two primes. In the above-mentioned context of character fields of values, it is natural to consider further restrictions on the values of such ′ -degree characters.
In Theorem A, we show that every finite group of order divisible by 2 or q possesses a nontrivial irreducible character of {2, q} � -degree with field of values contained in ℚ(e 2i ∕q ) , a surprising result that generalizes both [23] and [24,Theorem B] in the fashion of [8].
Theorem A Let G be a finite group, let q be a prime and write = {2, q} . Then, G possesses a nontrivial ′ -degree irreducible character with field of values contained in ℚ(e 2 i∕q ) if, and only if, gcd(|G|, 2q) > 1.
The obvious problem suggested by Theorem A is to try to understand when the irreducible character it identifies can be chosen to be rational, that is, when such character can be chosen to have only rational values. In other words, for a group G of even order and an odd prime q, we would like to characterize when G has a ′ -degree rational character, where = {2, q} . This is not always the case, in contrast to what happens if we allow small cyclotomic field extensions of ℚ as fields of values, as described by Theorem A. For example, the only rational linear character of 4 is the trivial one. A complete answer to this problem appears difficult to achieve and at the time of this writing, we do not know what form such a classification would take. However, in the case where G is a solvable group (or an alternating group, see Theorem D), we can completely solve this problem.
Theorem B Let G be a solvable group, q be a prime and set = {2, q} . Then, G admits a nontrivial rational irreducible character of ′ -degree if, and only if, H∕H � has even order, where H ∈ Hall (G).
We care to remark that the statement of Theorem B does not hold outside solvable groups, as shown by 5 and = {2, 3} (see Remark 5.11).
Our proof of Theorem A relies on the Classification of the Finite Simple Groups. In fact, for alternating groups and generic groups of Lie type, the arguments naturally extend from a pair {2, q} of primes to any pair {p, q} . Hence we obtain Theorem A as a corollary of the following statement, which classifies finite groups admitting a ′ -degree character with values in certain cyclotomic extensions of ℚ , for any set consisting of two primes. Note that the seemingly random exceptions in Theorem C suggest that the use of the CFSG is perhaps unavoidable in the present context. From now on, we will use ℚ( ) to denote the field of values of a character .
Theorem C Let G be a finite group and = {p, q} be a set of primes such that either p or q divides |G|. Assume that: Then, G possesses a nontrivial irreducible character of ′ -degree such that ℚ( ) ⊆ ℚ(e 2 i∕p ) or ℚ( ) ⊆ ℚ(e 2 i∕q ).
Theorem C has been used in [11] to obtain a lower bound for the number of almost p -rational irreducible characters of p ′ -degree in a finite group G. A character is said to be almost p-rational if ℚ( ) ⊆ ℚ(e 2 i∕n ) for some nonnegative integer n with p-part at most p. The aforementioned refinement of the McKay conjecture [20,Conjecture A] would imply that the number of almost p-rational irreducible characters of p ′ -degree of G is at least the number of conjugacy classes in the group G (P)∕Φ(P) , where P ∈ Syl p (G) and Φ(P) is its Frattini subgroup. Therefore, we expect that any finite group of order divisible by p has many almost p-rational irreducible characters of p ′ -degree. Notice that Theorems A and C are consistent with this new consequence of the Galois refinement of the McKay conjecture, as the ′ -degree characters identified by them are almost p-rational regardless of the prime q.
Finally, and as briefly mentioned before stating Theorem B, we are able to completely determine which simple alternating groups admit a nontrivial rational irreducible character of ′ -degree, for any set consisting of exactly two primes. We will write Irr � (G) to denote the set of irreducible characters of G of ′ -degree.
Theorem D Let n ≥ 5 be a natural number and let p, q be distinct primes. Let = {p, q} . The alternating group n admits a nontrivial rational irreducible character of ′ -degree for all those n ∈ ℕ that do not satisfy any of the following conditions (up to possibly interchanging the primes p and q).
This paper is structured as follows: In Sect. 2, we prove Theorems A and C assuming Theorem 2.1 on finite simple groups. In Sect. 3, we prove Theorem D, which in particular yields the alternating group case of Theorem 2.1. In Sect. 4, we prove Theorem 2.1 for sporadic groups and simple groups of Lie type, thus completing the proof of Theorem 2.1. Finally, we prove Theorem B in Sect. 5.

Proofs of Theorems A and C
Given a character of a finite group and a field extension F of ℚ , we say that is F -valued if ℚ( ) ⊆ F . Recall that is always ℚ(e 2 i∕|G| )-valued. In the special cases where F is the field of rational or real numbers, we will sometimes just say that is rational or real, respectively. In particular, rational characters are real. Moreover, given a prime number p, we say that is p -rational if ℚ( ) ⊆ ℚ(e 2 i∕n ) for some nonnegative integer n not divisible by p [13,Definition 6.29].
The aim of this section is to prove Theorems A and C of the introduction. In order to do so, we assume the following result on finite simple groups. This will be shown to hold in Sect. 3 and 4. We start with a lemma. Proof If the stabilizer in G of is G = M then by the Clifford correspondence [13, Theorem 6.11], we have G ∈ Irr(G) with ℚ( G ) ⊆ ℚ( ) ⊆ ℚ(e 2 i∕p ) , as required. Therefore, we may assume that is G-invariant.
Note that is r-rational. It follows from [13,Theorem 6.30] that G has a unique r-rational irreducible constituent . Indeed, is extendible to , and hence ℚ( ) ⊆ ℚ( ).
For each ∈ Gal(ℚ( )∕ℚ( )) , obviously is also an r-rational character of G lying over . Therefore, by the uniqueness of , we have = . Then, is Gal(ℚ( )∕ℚ( )) -fixed, which implies that ℚ( ) ⊆ ℚ( ) . We have shown that ℚ( ) = ℚ( ) ⊆ ℚ(e 2 i∕p ) , as desired. ◻ We will often use the following classic result of Burnside. We include its elementary proof to emphasize the difference between reality and rationality of characters mentioned in the introduction. If |G ∶ G � | is even, then G has a normal subgroup H of index 2 and the only nontrivial irreducible character of G/H is rational. It follows that G has a nontrivial real character. Otherwise, the sum of the squares of the degrees of nonlinear characters of G is odd. Then, the action of the complex conjugation on characters must leave some nonlinear irreducible character of G invariant. Therefore is a nontrivial real irreducible character of G.
Let G = G 0 ⊳ G 1 ⊳ ⋯ ⊳ G n = 1 be a composition series of G and let 0 ≤ k ≤ n − 1 be the smallest such that G k ∕G k+1 is either nonabelian simple or cyclic of order 2, p, or q. In particular, G i ∕G i+1 is cyclic of order coprime to 2pq for every i < k.
If G k ∕G k+1 is cyclic of order 2, p, or q, then obviously G k ∕G k+1 has a nontrivial irreducible character of ′ -degree such that ℚ( ) ⊆ ℚ(e 2 i∕p ) or ℚ( ) ⊆ ℚ(e 2 i∕q ) . On the other hand, when G k ∕G k+1 =∶ S is nonabelian simple, Theorem 2.1 implies that there exists Viewing the above as a character of G k , we now know that G k possesses a nontrivial irreducible character k of ′ -degree such that ℚ( k ) ⊆ ℚ(e 2 i∕p ) or ℚ( k ) ⊆ ℚ(e 2 i∕q ) . Using Lemma 2.2, we obtain k−1 ∈ Irr(G k−1 ) lying over k with ℚ( k−1 ) ⊆ ℚ(e 2 i∕p ) or ℚ( k−1 ) ⊆ ℚ(e 2 i∕q ) . Moreover, following the proof of Lemma 2.2, we see that Repeating this process k times, we can produce a nontrivial irreducible character ∶= 0 of ′ -degree such that ℚ( ) ⊆ ℚ(e 2 i∕p ) or ℚ( ) ⊆ ℚ(e 2 i∕q ) . ◻ Theorems A and C follow immediately from Theorem 2.4.

Alternating groups
The aim of this section is to prove Theorem 2.1 for alternating groups. In order to do so, we completely describe alternating groups possessing a rational-valued ′ -degree character. This is done by proving Theorem D of the introduction, which might be of independent interest. We begin by recalling that irreducible characters of the symmetric group n are labeled by partitions of n [17,Chapter 2]. We denote by the irreducible character of n corresponding to the partition of n. We will sometimes use the notation ⊢ n to mean that is a partition of n. Similarly we will write ⊢ p ′ n to say that (1) is coprime to p. Given a partition of n, we denote by ′ its conjugate. If ≠ ′ then ( ) n ∈ Irr( n ) . On the other hand, if = � then ( ) n = + g for some ∈ Irr( n ) and g ∈ n ∖ n .
Assuming that the reader is familiar with the basic combinatorial concepts involved in the representation theory of symmetric groups (as explained for instance in [27, Chapter 1]), we recall some important facts that will play a crucial role in our proofs. Given ⊢ n and i, j ∈ ℕ we denote by h ij ( ) the length of the hook of corresponding to node (i, j). For e ∈ ℕ , we let H e ( ) be the set consisting of all those nodes (i, j) of such that e divides h ij ( ) . Moreover, we let C e ( ) denote the e-core of .
For any natural number m, we denote by p (m) the exponent of the maximal power of p dividing m. The following lemma follows from [27, Proposition 6.4].

Lemma 3.1 Let p be a prime and let n be a natural number with p-adic expansion
A consequence of Lemma 3.1 is highlighted by the following statement. A second useful consequence of [27,Proposition 6.4] is stated in the following lemma.
We conclude this brief background summary by recalling a well-known fact on cyclotomic extensions of the rational numbers [1, Lemma 1.2.1].

Lemma 3.4 If p is an odd prime number, then
We are now ready to prove the main result of this section, which is Theorem D in the introduction.
Proof of Theorem D Assume first that both primes p and q divide the order of n . Equivalently, we have p, q ≤ n . Let n = ∑ t i=1 a i p m i = ∑ r j=1 b j q k j be the p-adic and, respectively, q-adic expansions of n. Here m 1 > m 2 > ⋯ > m t ≥ 0 and k 1 > k 2 > ⋯ k r ≥ 0 . Without loss of generality, we can assume that b 1 q k 1 < a 1 p m 1 . We consider ∈ P(n) to be defined by: As done in the proof of [8, Theorem 2.8], we observe that ∈ Irr � ( n ) and that (1) ≠ 1 unless n = a 1 p m 1 = b 1 q k 1 + 1 . We also claim that ≠ ′ . This follows by observing that = � would imply that Both these situations cannot occur. We conclude that ∶= ( ) n ∈ Irr � ( n ) and that ℚ( ) = ℚ. Let us now consider the case where n = ap m = bq k + 1 , for some m, k ∈ ℕ , some Let us now study the situation where a, b ∈ {1, 2} . Since ap m = bq k + 1 we observe that the only cases to consider are (a, b) ∈ {(1, 1), (1, 2), (2, 1)}.
• If (a, b) = (1, 2) then n = p m = 2q k + 1 and hence p ≠ 2 . Since 2 = b ≤ q − 1 we also have that q ≠ 2 . By Lemma 3.2 we deduce that ∈ Irr p � ( n ) if and only if = (d, 1 n−d ) is a hook partition. Moreover, since q is odd, again from Lemma 3.1 we observe that the only hook partitions of n that label characters of n of degree coprime to q are (n), (1 n ) and = (1 + q k , 1 q k ) = � . We also observe that m must be odd in this situation, as p m = 2q k + 1 ≡ 3 mod 4 . It follows that n admits exactly two distinct nontrivial irreducible characters of ′ -degree: the two irreducible constituents 1 , 2 of ( ) n . By [17, 2.5.13] we observe that their fields of values are equal to ℚ( √ −p m ) and strictly contain ℚ . Moreover, since m is odd then p ≡ p m = 2q k + 1 ≡ 3 mod 4 . Hence using Lemma 3.4 we observe that for all i ∈ {1, 2} we have The situation is similar to the one described above. Using Lemma 3.2 we notice that the nonlinear irreducible characters of n of degree coprime to q are labeled by all partitions such that h 11 ( ) = q k and h 22 ( ) = 1 . Since 2 = a ≤ p − 1 , we have that p ≠ 2 . Therefore, Lemma 3.1 implies that the only partition that labels a nonlinear irreducible character of n of degree coprime to p and to q is = (p m , 2, 1 p m −2 ) = � . As before we deduce that n admits exactly two distinct nontrivial irreducible characters of ′ -degree: the two irreducible constituents 1 and 2 of ( ) n . By [17, 2.5.13] we observe that for all i ∈ {1, 2} we have that ℚ( i ) = ℚ( √ q k ) . It follows that for any i ∈ {1, 2} , ℚ( i ) strictly contains ℚ if and only if k is odd. In this case, for all i ∈ {1, 2} we have that ℚ( i ) = ℚ( √ q) . Moreover, since p ≠ 2 then q ≡ 1 mod 4 . Therefore ℚ( i ) ⊆ ℚ(e 2 i∕q ) , by Lemma 3.4.
• If (a, b) = (1, 1) then exactly one of p or q is equal to 2. If q = 2 then n = p m = 2 k + 1 . By Lemma 3.2 we deduce that ∈ Irr p � ( n ) if and only if = (d, 1 n−d ) is a hook partition. Lemma 3.2 shows that (n), (1 n ) are the only hook partitions labeling an odd-degree character of n . Moreover, using Lemma 3.3 we observe that the partition = (1 + 2 k−1 , 1 2 k−1 ) = � , is the only hook partition of n such that 2 ( (1)) = 1 . We deduce that the two irreducible constituents 1 and 2 of ( ) n are the only nontrivial irreducible characters of ′ -degree of n . By [17, 2.5.13] we observe that ( 2 ) ) strictly contain ℚ if and only if m is odd. In this case, p ≡ p m ≡ 2 k + 1 ≡ 1 mod 4 . Therefore Lemma 3.4 implies that for all i ∈ {1, 2} we have that If p = 2 then n = 2 m = q k + 1 and we notice that k is necessarily odd. Moreover, Lemma 3.2 implies that the only {2, q} � -degree irreducible characters of n are the linear ones. On the other hand, Lemma 3.3 shows that = (2 m−1 , 2, 1 2 m−1 −2 ) = � is the only partition labeling a q ′ -degree irreducible character of n such that 2 ( (1)) = 1 . Arguing as above, we deduce that n admits exactly two distinct nontrivial irreducible characters of ′ -degree: the two irreducible constituents 1 and 2 of ( ) n . By [17, 2.5.13] we observe that for all To complete the proof we need to treat the easier case where pq does not divide | n | = n!∕2 . In this setting, we just need to show that there exists a rational-valued ′ -degree irreducible character of n (as conditions (i) and (ii) of the statement of Theorem D cannot be satisfied).
If p, q > n , then every irreducible character of n has ′ -degree. Since n ≥ 5 , we have (n − 1, 1) ≠ (n − 1, 1) � and therefore = ( (n−1,1) ) n is a nontrivial irreducible character of ′ -degree of n such that ℚ( ) = ℚ . Otherwise, up to possibly interchanging p and q we can assume that p ≤ n < q . In this case, every irreducible character of n has degree coprime to q. Let n = a + pw for some 0 ≤ a ≤ p − 1 and w ≥ 1 . Consider = (n − (a + 1), a + 1) ∈ P(n) . Since n ≥ 5 we notice that necessarily ≠ ′ . Moreover, Lemma 3.1 implies that p does not divide (1) . It follows that = ( ) n is a nontrivial irreducible character of ′ -degree of n such that ℚ( ) = ℚ . ◻ A straightforward consequence of Theorem D is that Theorem 2.1 holds for alternating groups. Proof If n does not satisfy conditions (i) and (ii) of Theorem D, then n has a nontrivial rational character. If n satisfies condition (i), then the proof of Theorem D shows that there exists ∈ Irr � ( n ) such that (1) > 1 and such that ℚ( ) ⊆ ℚ(e 2 i∕p ) . On the other hand, if n satisfies condition (ii), we have shown in the proof of Theorem D that there exists ∈ Irr � ( n ) such that (1) > 1 and such that ℚ( ) ⊆ ℚ(e 2 i∕q ) . ◻

Simple groups of Lie type
In this section, we prove Theorem 2.1 for simple groups of Lie type and sporadic simple groups. The following reduces us to the case of simple groups of Lie type with nonexceptional Schur multipliers. The list of finite simple groups with exceptional Schur multipliers is available in [10, Proof This can be confirmed using GAP and the Atlas [4,32]. In particular, the character tables for the groups under consideration are available in the GAP Character Table Library, and we make use of the +Conductor+ command in GAP, which returns the smallest natural number m for which a character in a stored character table takes its values in ℚ(e 2 i∕m ) . ◻ When S is a simple group of Lie type, the required character ∈ Irr(S) of ′ -degree we produce will be a semisimple character. Let us recall some brief background on these characters from [2,5,7]. (We refer the reader in general to these references for more on the character theory of groups of Lie type.) Let G be a connected reductive algebraic group in characteristic p and F a Frobenius endomorphism of G . For each rational maximal torus T of G and character ∈ Irr(T F ) , Deligne-Lusztig's twisted induction R G T is used to define the Deligne-Lusztig character R G T ( ) . Let G * be an algebraic group with a Frobenius endomorphism F * such that (G, F) is dual to (G * , F * ) . Set G ∶= G F and G * ∶= (G * ) F * . Recall that if (T, ) is G-conjugate to (T � , � ) , then R G T ( ) = R G T � ( � ) . Moreover, by [5, Proposition 13.13], the G-conjugacy classes of pairs (T, ) are in one-to-one correspondence with the G * -conjugacy classes of pairs (T * , s) , where s is a semisimple element of G * and T * is a rational maximal torus containing s. Due to this correspondence, we can use the notation R G T * (s) for R G T ( ) . For each conjugacy class (s) of semisimple elements in G * such that G * (s) is connected, one can define a so-called semisimple character of G as follows: where W(s) is the Weyl group of G * (s) , T * w is a torus of G * of type w, and G = ±1 , depending on whether the relative rank of G is even or odd, see [5,Definition 14.40]. Moreover, (s) is irreducible and where we recall that p is the defining characteristic of G and n p ′ denotes the p ′ -part of a positive integer n. Lemma 4.2 With the notation as above, let s ∈ G * be a semisimple element such that G * (s) is connected. Suppose s has order k and ∈ Gal(ℚ(e 2 i∕|G| )∕ℚ) satisfies ( ) = m for every kth root of unity , where m is an integer relatively prime to k. Then, (s) = (s m ) . In particular,ℚ( (s) ) ⊆ ℚ(e 2 i∕k ).
Proof Let T * be a rational maximal torus of G * containing s. Let T be a rational maximal torus of G and ∈ Irr(T F ) such that the G-conjugacy class of (T, ) corresponds to the G * -conjugacy class of (T * , s) under the correspondence described above. Then, (T, m ) corresponds to (T * , s m ) , and the multiplicative order of in the group Irr(T F ) is the same as the order of s. Therefore, the values of are in ℚ(e 2 i∕k ).
We recall the character formula for R G T � ( ) , which we simplify as R T ′ , : where t is semisimple, u is unipotent, and g = tu = ut is the Jordan decomposition of g ∈ G . Also, 0 G (t) is the connected component of G (t) and Q is not connected, we may embed G into another connected reductive group G satisfying (G) is connected, via a so-called regular embedding ∶ G ↪G . We record here some of the properties of regular embeddings. For proofs and a nice detailed discussion, we refer the reader to [7, Section 1.7]. In this situation, may be chosen so that, identifying G with its image (G) in G , the Frobenius endomorphism F is naturally the restriction of a Frobenius endomorphism on G , which we will also denote by F. Then, writing G ∶=G F , we have G ⊲G , the quotient group G ∕G is abelian of order prime to p, and the restriction ResG G ( ) to G of any ∈ Irr(G) is multiplicity-free.
Further, the regular embedding induces a dual map * ∶G * → G * that maps G * ∶= (G * ) F * onto G * and whose kernel is central and F * -fixed. With this in place, following the treatment in [7, Corollary 2.6.18], we may define the semisimple characters of G corresponding to a given semisimple s ∈ G * to be the irreducible constituents of the restriction ResG G ( (s) ) of a semisimple character (s) of G , where s is a semisimple element of G * such that * (s) = s . Fixing s to be one such semisimple character of G corresponding to s, we again have Further, the set Irr(G) may be partitioned into subsets known as rational Lusztig series, denoted E(G, s) , which are indexed by G * -conjugacy classes of semisimple elements s ∈ G * (see, e.g., [5,Proposition 14.41]). In particular, the semisimple characters of G corresponding to s lie in the set E(G, s) . When G * (s) is connected, (s) is the unique semisimple character in E(G, s) . To avoid confusion, we will use the notation (s) only in the case G * (s) is connected.
In the case G * (s) is disconnected, and hence Lemma 4.2 does not apply, the following, extracted from [28], will be useful. Here, we write E(G, s) for the set { | ∈ E(G, s)}. We remark that the same proof as [28,Lemma 3.4], which is analogous to that of Lemma 4.2, yields the corresponding statement of Lemma 4.3(i) for arbitrary |s|, but we will not need this here. Now, if S is a simple group of Lie type not isomorphic to an alternating group or one listed in Proposition 4.1, then S = G∕ (G) for G = G F , where G is simple of simply connected type. The following will be used in conjunction with Lemma 4.3 in many cases in which G * (s) is not connected. Although it can be extracted from the proofs of [8, Lemma 3.3 and Theorem 3.5], we rewrite the proof for the convenience of the reader. Proof We may assume S is not one of the groups listed in Proposition 4.1 nor isomorphic to an alternating group. Further, thanks to [23,24], we may assume that p ≠ q. Let S be of the form G∕ (G) , where G = G F is the set of fixed points of a simple connected reductive algebraic group of simply connected type defined in characteristic r, under a Frobenius endomorphism F. Note that the Steinberg character St G of G has degree 1 3 a power of r, is rational-valued, and is trivial on (G) . Hence, we may assume that r = p is one of the primes in .
Let G be of type A n−1 . Then, G = SL n and G = GL n , and we may write n = a 1 q + ⋯ a t q t with 0 ≤ a i < q for 1 ≤ i ≤ t . We write G = GL n (p a ) and G = SL n (p a ) , where ∈ {±1} and = 1 gives the untwisted version SL n (p a ) , and = −1 gives the twisted version SU n (p a ) . Recall that G * (s) is connected for any semisimple s ∈G * . Further, note that and G * ≅G∕ (G) = PGL n (p a ) . Throughout, we will make these identifications. Let Q ∈ Syl q (G) . Then, by [3,34] First, assume that n is not a power of q. Let s � ∈ (Q k ) have order q. If n ≠ 2q k , define s ∈ (Q) to be of the form diag(s � , I n−q k ) . If q | (p a − ) , then s ′ may further be chosen to be of the form I q k ∈ (GL q k (p a )) , where ∈ C p a − ≤ × p 2a has order q. Then, det(s) = det(s � ) = q k = 1 . Otherwise, q ∤ |G∕G| , so Q ≤ G . In either case, s ∈ G = (G * ) � , so the corresponding semisimple character (s) of G is trivial on (G) , by [25,Lemma 4.4]. If q is odd and n = 2q k , we may instead let s ∈ (Q) be of the form diag( I q k , −1 I q k ) if q | (p a − ) and diag(s � , I q k ) if q ∤ (p a − ) , and we again see that s ∈ G . Further, since the conjugacy classes of semisimple elements of G are determined by their eigenvalues, we see s is not G -conjugate to sz for any nontrivial z ∈ (G * ) . But the irreducible characters of G ∕G are in bijection with elements of (G * ) , and (� s) ⊗ � z = (� sz) for ẑ ∈ Irr(G∕G) corresponding to z ∈ (G * ) (see [5, 13.30]). Hence (s) is also irreducible when restricted to G. By Lemma 4.2, (s) has values in ℚ(e 2 i∕q ) , so this yields a member of Irr(S) with degree prime to both p and q and with values in ℚ(e 2 i∕q ) , as desired. Now assume that n = q k . Then, [8,Lemma 3.4] yields that any irreducible character of G with degree prime to q is trivial on the center, which has size gcd(n, p a − ) . Hence it suffices to show there exists a member of Irr(G) with degree prime to p and to q whose values lie in ℚ(e 2 i∕p ) or ℚ(e 2 i∕q ).
Let s be as in Lemma 4.4, where we let Q =Q (G)∕ (G) ∈ Syl q (G * ) , and let s ∈Q (G) be such that s (G) = s . Then, notice s q ∈ (G) . Let = e 2 i∕|s| , so that the semisimple character ̃ (s) of G corresponding to s takes its values in ℚ( ) , by Lemma 4.2, and lies over the {p, q} � -degree character s of G. Let ∈ Gal(ℚ( )∕ℚ(e 2 i∕q )) . Then, maps to m for some m with gcd(m, |s|) = 1 . Further, m ≡ 1 (mod q) , since fixes qth roots of unity. In particular, s m =sz for some z ∈ (G) . Then, using Lemma 4.2, we have ̃ (s) =̃ (s m ) =̃ (sz) , and hence ̃ (s) also lies over s . In particular, Res ( � (� s) ) . So, ResG G (̃ (s) ) is fixed by each such , and therefore has values in ℚ(e 2 i∕q ).
Hence we may assume G is not of type A n−1 , and therefore | (G)| is a power of q. Let s be the character constructed in Lemma 4.4. In this case, note that either q = 2 or (G, q) = (E 6 , 3) . In the latter case, G is the simply connected type group E 6 (p a ) sc , where ∈ {±1} , = 1 corresponds to the untwisted version, and = −1 corresponds to the twisted version. Using [31, Theorem 1.8 and Lemma 2.6], we see that any irreducible character of G takes integer values on unipotent elements, and hence Lemma 4.3 yields that s is fixed by any Galois automorphism that fixes the field ℚ(e 2 i∕3 ) , and hence has values in ℚ(e 2 i∕3 ).
We may therefore take q = 2 , p odd, and G to be of type B n , C n , D n , or E 7 . Here s 2 = 1 , and by Lemma 3.4, it suffices to show that s takes values in ℚ( √ p) . The data available in CHEVIE and [18] yield that the odd-degree characters of 3 D 4 (p a ) and E 7 (p a ) are all rational-valued, and hence we assume S is B n (p a ) with n ≥ 2 , C n (p a ) with n ≥ 3 , D n (p a ) with n ≥ 4 , or 2 D n (p a ) with n ≥ 4. If

Rational characters of ′ -degree in solvable groups
In this section, we prove Theorem B. Namely, we characterize when a solvable group G has a ′ -degree rational character, where = {2, q} is a pair of primes. We first note that if G has a normal Hall {2, q}-subgroup, then the solution to the characterization problem is pretty simple in Lemma 5.1. We thank G. Navarro for pointing out a simplified version of a previous argument.
In what follows, we assume that the reader is familiar with the theory of characters and normal subgroups as in [13,Chapter 6], for instance. Proof Notice that to prove both implications, we may assume that H is abelian. If H has order divisible by p, then let H ≠ ∈ Irr(H) be linear with o( ) equal to p. By [13, Corollary 6.27], let ̂∈ Irr(G ) be the only extension of such that o(̂) = o( ) . In particular, ℚ(̂) = ℚ(e 2 i∕p ) and hence G ≠ = (̂) G ∈ Irr(G) (by [13,Theorem 6.11]) has values in ℚ(e 2 i∕p ) . Since G/H is a ′ -group, it follows that has ′ -degree, as wanted.
Suppose now that G ≠ ∈ Irr(G) has ′ -degree, ℚ( ) ⊆ ℚ(e 2 i∕p ) , and p does not divide |H|. If p = 2 , then is a nontrivial rational irreducible character of an odd-order group. This would contradict Theorem 2.3. If p > 2 , then |H| = q b and since G/H is a ′ -group, then G is a p ′ -group. In particular ℚ( ) ⊆ ℚ(e 2 i∕p ) ∩ ℚ(e 2 i∕|G| ) = ℚ . Since G/H is an odd-order group by hypothesis, and q > p > 2 also by hypothesis, then we get again a contradiction with Theorem 2.3.
Finally, let ∈ Irr(H) be a constituent of H . Then, is linear and nontrivial. Indeed, if = H , then would be a rational irreducible character of G/H of odd order, hence = G by Burnide's theorem, contradicting one of the assumptions on . We consider G * = {g ∈ G | g = for some ∈ Gal(ℚ( )∕ℚ)} ≤ G , sometimes known as the semiinertia group of in G. By [22,Lemma 2.3] the map given by g ↦ , whenever g ∈ G * and g = , defines a monomorphism G * ∕G → Gal(ℚ( )∕ℚ) . Write E = ℚ( ) . Since is a linear character of a -group, then E is obtained by adjoining a root of unity of order o( ) = p a q b to ℚ , where a, b ≥ 0 are not both zero. Write = e 2 i∕p . We claim that ℚ( ) ⊆ E . Otherwise, o( ) = q b with b ≥ 1 , and E ∩ ℚ( ) = ℚ . Take any ∈ Gal(E∕ℚ) and extend it to ∈ Gal(E( )∕ℚ( )) by elementary Galois theory. Since = , then = = g for some g ∈ G . Note that g ∈ G * . In particular, contradicts the oddness hypothesis on |G/H| as 2 ≤ p < q . Now we know that ℚ( ) ⊆ E . The above argument actually proves that Gal(E∕ℚ( )) injects into G * ∕G , since The fact that |G/H|, hence |G * ∕G | , is odd and coprime to p forces a = 1 and b = 0 , that is, We are working in slightly more generality than needed in this section. We note that if p = 2 , then the condition on the order of G/H is trivially satisfied in Lemma 5.1. However, it is a necessary condition in general: if p is an odd prime, then the group G = p ⋊ p−1 where the action is faithful has a ′ -degree rational irreducible character for every = {p, q} with q a divisor of p − 1.
For an arbitrary group G and a set of primes , we will denote by X � , p (G) the set of ′ -degree irreducible characters of G with values in ℚ(e 2 i∕p ) . If G is -separable, then the set X � , p (G) consists entirely of monomial characters, given that | G (H)∕H| is odd for H ∈ Hall (G) . In [26] this fact is proven in the case where consists of a single prime, but it generalizes to arbitrary , see Theorem 5.4. Recall that by work of P. Hall and S. A. Čunhihin, Hall -subgroups in -separable groups behave like Sylow subgroups.

Lemma 5.2
Let G be -separable, H ∈ Hall (G) , and ∈ Irr � (G) . If M ⊲ G , then M contains some H-invariant irreducible constituent and any two of them are G (H)-conjugate.
Proof Let ∈ Irr(M) be a constituent of M . By the Clifford correspondence [13, Theorem 6.11], |G ∶ G | is a ′ -number. Since G is -separable, then H ⊆ G x = G x for some x ∈ G , and x is an H-invariant constituent of M . Write = x . Suppose that y is also H-invariant. Then H and H y −1 are Hall -subgroups of G . In particular, H y −1 = H g for some g ∈ G . Note that gy ∈ G (H) and y = gy . ◻ The following is a nice application of Theorem 2.3 in the context of coprime group actions.  Proof We first prove the existence of a pair by induction on |G|. We may assume that H < G and G ′ > 1 since otherwise would be linear and there is nothing to prove. We may assume that ker( ) = 1 by working in G∕ ker( ).
Let M ⊲ G . We claim that if for some H-invariant irreducible constituent of M , also lies under , then = . Indeed, if and are H-invariant irreducible constituents of M , then = x with x ∈ G (H) by Lemma 5.2. Write N = G (H) . Note that x 2 ∈ N . Since |N : H| is odd, then ⟨xH⟩ = ⟨x 2 H⟩ , so x ∈ N and the claim follows.
We may assume that and K is a ′ -group. Hence lies under and by the paragraph above = . Note that K (H) ≅ KH (H)∕H can be seen as a subgroup of G (H)∕H , and hence K (H) has odd order. By Lemma 5.3, we obtain that = K , contradicting the fact that is faithful.
Let M = (G) . We show that M > 1 is abelian. By [14,Theorem 3.21] we have that M > 1 . Let ∈ Irr(M) be under . Since has ′ -degree, we have that is linear. Then M ′ is contained in the kernel of every G-conjugate of . Since is faithful, we conclude that M � = 1.
Let L be a minimal normal subgroup of G contained in G ′ . Since G is -separable and � (G) = 1 , then L ⊆ M is q-elementary abelian for some q ∈ . Let ∈ Irr(L) be H-invariant lying under . Note that is linear. We show that ℚ( ) ⊆ ℚ(e 2 i∕p ) . If q ≠ p , then ℚ( L ) = ℚ(e 2 i∕p ) ∩ ℚ(e 2 i∕|L| ) = ℚ . Hence and lie under , and by the second paragraph of this proof = . In particular ℚ( ) = ℚ ⊆ ℚ(e 2 i∕p ) (in this case q = 2 necessarily). If L is p-elementary abelian, then ℚ( ) = ℚ(e 2 i∕p ).
Next we show that extends to its stabilizer G and consequently G < G . In order to show that extends to G , it is enough to show it extends to every Sylow subgroup of G ∕L by [13,Theorem 6.26]. Let R∕L ∈ Syl r (G ∕L) . If r ≠ q , then extends to R because L is a q-group and [13, Corollary 6.27]. If r = q , then R is a q-group. Since has q ′ -degree, then R has some linear constituent and hence extends to R. Note that if G = G , and ∈ Irr(G) is an extension of then L ⊆ G � ⊆ ker( ) . Then = L , a contradiction with ker( ) = 1.
Let ∈ Irr(G ) lying over be the Clifford correspondent of as in [13, Theorem 6.11]. Since both and have values in ℚ(e 2 i∕p ) , so does . We have H ⊆ G < G , and by induction there exists a pair (U, ) of the desired kind inducing , hence also inducing .
For the uniqueness up to G-conjugacy, see the second part of the proof of [33, Theorem 2.6]. ◻ The fact that under the hypothesis of Theorem 5.4, the set X � ,p (G) consists entirely of monomial characters allows us to construct a map between X � ,p (G) and X � ,p ( G (H)) as in [12,Theorem C]. In order to prove that such map is bijective, we will need a -version of [16,Theorem 3.3], which relies on -versions of [16, Theorem 2.1 and Corollary 2.2]. All these versions hold without restrictions on the set . Their proofs can be obtained by just mimicking the proofs in [16] and using the following easy observation. Proof Since extends to G, note that H � ⊆ H ∩ G � ⊆ ker( ) . In particular, seen as a character of H∕H ∩ G � ≅ HG � ∕G � ⊲ G∕G � extends to G∕G � . By [13,Corollary 6.27], extends to some ∈ Irr(G∕G � ) with o( ) = o( ) and the statement follows. Proof We are basically going to follow the proof of [33,Theorem 2.13]. Given ∈ X � ,p (G) , the existence of the a pair (U, ) as in the statement is guaranteed by Theorem 5.4. Set Ω( ) = ( U∩N ) N . By [12, Lemma 2.3.(a)] we have that Ω( ) ∈ Irr(N) , so in particular Ω( ) ∈ X � ,p (N) . Since any other inducing pair for is G-conjugate to (U, ) and Ω ∶ X � ,p (G) → X � ,p (N)