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A method for obtaining very precise results along the lines of the Hilbert Irreducibility Theorem is described and then applied to a special case. In addition, the relationship of the irreducibility theorem to other tools of diophantine analysis is investigated. In particular, we give a proof of the irreducibility theorem that uses only Noether's lemma and the fact that an absolutely irreducible curve has a rational point over a finite field of large order.

We prove here the following theorems: A. If k is a denumerable Hilbertian field then for almost all ( σ 1 , … , σ e ) ∈ G ( k s / k ) e ({\sigma _1}, \ldots ,{\sigma _e}) \in \mathcal {G}{({k_s}/k)^e} the fixed field of { σ 1 , … , σ e } , k s ( σ 1 , … , σ e ) \{ {\sigma _1}, \ldots ,{\sigma _e}\} ,{k_s}({\sigma _1}, \ldots ,{\sigma _e}) , has the following property: For any non-void absolutely irreducible variety V defined over k s ( σ 1 , … , σ e ) {k_s}({\sigma _1}, \ldots ,{\sigma _e}) the set of points of V rational over K is not empty. B. If E is an elementary statement about fields then the measure of the set of σ ∈ G ( Q ~ / Q ) \sigma \in \mathcal {G}(\tilde Q/Q) (Q is the field of rational numbers) for which E holds in Q ~ ( σ ) \tilde Q(\sigma ) is equal to the Dirichlet density of the set of primes p for which E holds in the field F p {F_p} of p elements.

Let G be a finite group and p a rational prime. If G is a simple Lie type group of characteristic p then G has the following property. G has an absolutely irreducible character )~ whose degree is the order of a p-Sylow subgroup of G, and all other characters are in the principal p-block, Bo (P). (Z is called the Steinberg character. See [4, 5, 9] and [12].) For simple groups G, this property seems to hold only if G has Lie type of characteristic p. The most general conjecture that can be made from this observation, namely that the above property about the characters of G forces G to be of Lie type at characteristic p, seems too difficult at present. In fact to prove anything in this direction, I have had to make arithemtical assumptions about G as well as assumptions about a subgroup which ultimately corresponds to the normalizer of a torus. In Section 4 1 will indicate why such assumptions may be inevitable.

Asymptotic formulas are derived for the distribution of absolutely irreducible polynomials in two indeterminates over finite fields. A pair of inversion formulas is presented which yield exact formulas relating the distributions of irreducible and absolutely irreducible polynomials.

There are some simple facts which distinguish Lie-algebras over fields of prime characteristic from Lie-algebras over fields of characteristic zero. These are(1) The degrees of the absolutely irreducible representations of a Lie-algebra of prime characteristic are bounded whereas, according to a theorem of H. Weyl, the degrees of the absolutely irreducible representations of a semi-simple Lie-algebra over a field of characteristic zero can be arbitrarily high.(2) For each Lie-algebra of prime characteristic there are indecomposable representations which are not irreducible, whereas every indecomposable representation of a semi-simple Liealgebra over a field of characteristic zero is irreducible (cf.[4]).(3) The quotient ring of the embedding algebra of a Lie-algebra over a field of prime characteristic is a division algebra of finite dimension over its center, whereas this is not the case for characteristic zero. (cf.[4]).(4) There are faithful fully reducible representations of every Lie-algebra of prime characteristic, whereas for characteristic zero only ring sums of semi-simple Lie-algebras and abelian Lie-algebras admit faithful fully reducible representations (cf.[6], [2], [4]).

(1) Let g : V r → W m ( m ≧ r ) g:{V^r} \to {W^m}(m \geqq r) be a morphism of nonsingular varieties over an algebraically closed field. Under certain conditions, one can define a cycle S i {S_i} on V V with Supp ⁡ ( S i ) = { x | dim k ( x ) ( Ω X / Y 1 ) ( x ) ≧ i } \operatorname {Supp} ({S_i}) = \{ x|{\dim _{k(x)}}(\Omega _{X/Y}^1)(x) \geqq i\} . The multiplicity of a component of S i {S_i} can be computed directly from local equations for g g . If V r ⊂ P n {V^r} \subset {P^n} , and if g : V → P m g:V \to {P^m} is induced by projection from a suitable linear subspace of P n {P^n} , then S 1 {S_1} is c m − r + 1 ( N ⊗ O ( − 1 ) ) {c_{m - r + 1}}(N \otimes \mathcal {O}( - 1)) , up to rational equivalence, where N N is the normal bundle of V V in P n {P^n} . (2) Let f : X → S f:X \to S be a smooth projective morphism of noetherian schemes, where S S is connected, and the fibres of f f are absolutely irreducible r r -dimensional varieties. For a geometric point η : Spec ⁡ ( k ) → S \eta :\operatorname {Spec} (k) \to S , and a locally free sheaf E E on X X , let X η {X_\eta } be the corresponding geometric fibre, and E η {E_\eta } the sheaf induced on X η {X_\eta } . If E 1 , … , E m {E_1}, \ldots ,{E_m} are locally free sheaves on X X , and if i 1 + ⋯ + i m = r {i_1} + \cdots + {i_m} = r , then the degree of the zero-cycle c i 1 ( E 1 η ) ⋯ c i m ( E m η ) {c_{{i_1}}}({E_{1\eta }}) \cdots {c_{{i_m}}}({E_{m\eta }}) is independent of the choice of η \eta . (3) The results of (1) and (2) are used to study the behavior under specialization of a closed subvariety V ′ ⊂ P 2 r − 1 V’ \subset {P^{2r - 1}} which is the image under generic projection of a nonsingular V r ⊂ P n {V^r} \subset {P^n} .

Let X1, X2,…, Xk, denote k ≥ 2 indeterminates and let f(X1,…, Xk) be a homogeneous polynomial, in GF(pn) [X1,…, Xk], which is irreducible but not absolutely irreducible over GF(pn). Thus f is irreducible in GF(pn)[X1,…, Xk] but reducible in some GF(pnm) [X1,…, Xk], m > 1. For any polynomial h(X1,…, Xk) in GF(pnℓ)[X1,…, Xk, ℓ ≥ 1, let Npn(h) denote the number of (x1,…, xk) ∈ GF(pn)×…× GF(pn) such that (hx1…, xk) = 0.

Introduction. Among the valuation ideals in a polynomial ring = K [x, y] in two indeterminates, the ones of central importance are the simple valuation ideals, that is, the valuation ideals which are not products of two ideals different from 0, since every valuation ideal has a unique factorization into simple valuation ideals. The problem of the characterization of simple valuation ideals has been dealt with by Zariski, in the case that the field K is algebraically closed and of characteristic 0, in his paper, Polynomial ideals defined by infinitely near base points. There the problem is referred to the ring of holomorphic functions in x, y: *= K { x, y }, and a valuation ideal q in Z* is simple if and only if its general element is absolutely irreducible. If, however, only the characterization of the simple valuation ideals is desired, the notions of the general element of an ideal in D* and its absolute irreducibility are somewhat too strong for the problem set; although it should be stated that these notions are applied by Zariski to other topics not touched upon here. In this paper we treat the theory of simple valuation ideals by a more explicit and direct method and we also extend the theory to algebraically closed fields of arbitrary characteristic p 5 O. We characterize the simple v-ideals q in the sequence of zero-dimensional valuation ideals in ?, for a given valuation v, in terms of the value v(q) (under v) of q (that is, the least value assumed by elements of q) and of the least value greater than v(q) assumed by elements of ?). If q is not simple, an explicit factorization for q in terms of the two mentioned values is given Since in our treatment the field K is of arbitrary characteristic, Puiseux series expansions for valuations are not available. A corresponding tool is found in Theorem 6. There for a given valuation v, a certain (finite or infinite) sequence of polynomialsfi(x, y) is introduced. In the case that K is of characteristic 0, if v is given('), for example, by y=clxr(l)+c2xr(2)+ * * * , ciCK, r(i) =ri rational, with 0

Introduction. A well known theorem of Bertini-Enriques on reducible linear systems of Vr_-'s on an algebraic Vr (i.e., linear systems in which each element is a reducible Vri-) states that any such system, if free from fixed components, is composite with a pencil. The usual geometric proof of this theorem is based on another theorem of Bertini, to the effect that the general Vr_1 of a linear system cannot have multiple points outside the singular locus of the variety Vr and the base locus of the system. This geometric proof has been subsequently completed and presented by van der Waerden under an algebraic form [3]. In this paper we give a new proof of the theorem of Bertini on reducible linear systems and we also extend this theorem to irrational pencils, i.e., pencils of genus p >0. Our proof does not make use of the second theorem of Bertini just quoted. In the case of pencils (linear-or irrational), we first observe that a pencil { WI on Vr is defined by a field P of algebraic functions of one variable which is a subfield of the field z of rational functions on Vr. The whole proof is then essentially based on the simple remark that the pencil I W} is composite with another pencil { W }, defined by a field P, if and only if P is a subfield of P. This property is a straightforward consequence of the geometric definition of composite pencils. As a matter of fact we prefer to define composite pencils by this property. At any rate, it is then true that a pencil I W} is non-composite if and only if the corresponding field P is maximally algebraic in Z. In the light of this approach to the question, the theorem of Bertini on reducible pencils is almost a direct consequence of the well known fact that an irreducible algebraic variety V, over a ground field K, is absolutely irreducible if K is maximally algebraic in the field of rational functions on V. In the case of linear systems of dimension > 1 the proof is even simpler, provided use is made of a certain lemma (Lemma 5). This lemma is, however, of interest in itself. A sizable portion of the paper (Part I) is devoted to the development of the concept of a pencil and of the basic properties of pencils in the abstract case of an arbitrary ground field (of characteristic zero).

Introduction. Let K be an infinite perfect field of characteristic different from 2, let e3 be a quaternion division algebra over K, and let tt-denote the canonical involution of the first kind on Z. Let V be a finite-dimensional right vector space over Z. A quaternion skew-hermitian form H over Z is a sesquilinear form on VX V, i.e., H is a map from VX V to Z such that (i) H(x, Yl+Y2) =H(x, yi) +H(x, Y2) and H(x, yax) =H(x, y)a for all x, y, Y1, y2 in V and ai in ; (ii) H(x, y) = -H(y, x) for all x, y in V. Let {xl, . . *, x.) be a basis for V over Q. We say that H is nondegenerate if the reduced norm in Mn(Z) of the matrix (H(xi, xj)) is not zero. Associated to such a nondegenerate form H are 3 invariants, the dimension of V over 0, dime(V), the discriminant of H, b(H), and the Clifford algebra of H, G. Let G be a simply connected semisimple algebraic group (in some GL(m, X)) which is defined over K and let p:G-?GL(V/Z) be an absolutely irreducible representation of G defined over K into the group of all nonsingular Z-linear endomorphisms of V. We shall assume that there is a nondegenerate quaternion skew-hermitian form H on V which is invariant with respect to p (G). The purpose of this paper is to describe the Clifford algebra of the invariant form H in terms of p, G, and the Steinberg group associated to G. In a previous paper, we have described dimsz(V) and b(H) in such a way and have indicated how representations such as p arise [2, Theorem I.2 ]. The invariant -y(G) plays an important role in our study and so we recall some of its properties in ?1. In 2, we define the invariant Y using the representation p. Jacobson first constructed the Clifford algebra of a quaternion skew-hermitian form [4]. In this paper, however, we shall follow a method due to Satake [6]. We give some examples in ?3 with special emphasis on the case where G is absolutely simple.

It is customary to classify hypercomplex number systems as reducible and irreducible and, owing to the theorem that a reducible systemn can always be built up out of irreducible ones, the latter only are enumerated. In the followlowing paper, in which only systems with modulus are considered, the irreducible systems are further classified by means of their groups as " semireducible of the first kind" (" 1), "semireducible of the second kind" (? 2) and " absolutely irreducible " (? 3).

Let G be a finite group of order pam where p is prime and (p, m) = 1. Let Q be an algebraically closed field of characteristic p. We consider the group algebra A (G, Q) of G over Q; this group algebra possesses a radical which we denote by N(G, Q). We give an upper bound for the dimension of N(G, Q) and show that this upper bound is attained if and only if G has a normal p-Sylow subgroup. We follow closely the nomenclature of Brauer and Nesbitt [2]. Let G have p-regular classes C1, C2, * * *, Ck, where C1 consists of the identity element of G and C, contains g, elements (v = 1, 2, * * *, k). Let F1, F2, * * *, Fk, and U1, U2, * * *, Uk, be the absolutely irreducible and indecomposable representations of G over the field Q with modular characters 4)1, 42, * .., Ik, and t1, 12, , * * k, of degrees fi, f2, * * *, fk and u1, U2, . . , Uk respectively. We assume that F1 represents every element of G by the unit element of Q and that F, is the first and last irreducible constituent of U,, (K = 1, 2, * *, k). Denoting the dimension of N(G, Q) by dim N(G, Q), then we have the following upper bound due to Brauer and Nesbitt [2, p. 580]

The purpose of this paper is to determine explicitly, nondegenerate real symmetric bilinear forms invariant under a real absolutely irreducible representation of a real semisimple algebraic group, G G . If G G is split, we construct an extension G ∗ {G^ \ast } containing G G and those outer automorphisms of G G fixing the highest weight of the representation. The representation is then extended to G ∗ {G^ \ast } and the form is described in terms of the character of this extension. The case of a nonsplit algebraic group is then reduced to the above. The corresponding problem for representations by matrices over the real quaternion division algebra is also considered using similar methods.

Introduction.! As this paper is the expression of my work as a student under Prof. R. Brauer, I wish at the outset to acknowledge my deep indebtedness to my former teacher. For an associative algebra a, having a unit element and lying over a given field K, two regular representations T? and (E are defined. As we shall be concerned with the absolutely irreducible constituents of the regular representations we shall assume that K is algebraically closed. The decomposition of St and t is our first goal. Let 23 be a representation of an algebra a by linear transformations of a vector space V. If V is the direct sum of two vector spaces V, and V2, both invariant under A, then Z is decomposable. By adapting the co-ordinate system in V to the decomposition, we obtain Q3 in the form

Introduction. In 1964 C. W. Curtis classified the absolutely irreducible modular representations of a large class of groups, the so called finite groups of Lie type. This was done by finding in each irreducible module an element, called a weight element, which is an eigenvector for certain elements in the modular group algebra; proving that two irreducible modules are isomorphic if and only if the corresponding eigenvalues (the collection of such being called a weight) are equal; and finally by determining which weights are associated with irreducible modules, i.e. which weights actually occur. This paper is a continuation of Curtis' work. We construct the absolutely irreducible modular representations of a finite group of Lie type by finding weight elements in the modular group algebra which generate a full set of nonisomorphic minimal left ideals. In some work of Steinberg and Curtis (see [14] and [6]) these representations were constructed for the covering groups of the Chevalley groups using the representations of the associated modular Lie algebras. The discussion given here avoids the Lie algebras altogether. Instead we construct each irreducible submodule of certain induced modules by finding the required weight elements. We prove at the same time that these induced modules have multiplicity free socles. Some remarks are made about the related questions of degrees and block structure of these representations. The finite groups of Lie type whose representations were classified by Curtis in [7] were defined by a large number of axioms. These apparently consisted of basic properties possessed by all known examples of such groups, the Chevalley groups and variations thereon as defined by Steinberg [13], for example. A simplified axiom scheme is used here, the heart of which is the axioms for groups with BN pairs (Tits [16]). The groups discussed in [7] satisfy the simplified axioms. The disadvantage of introducing new axioms is that the classification described in the first paragraph of this introduction must be redone. The proofs in this reworking are always similar to Curtis' but rarely identical, for the new axioms are enough weaker that some of the properties Curtis used do not follow from them. The advantages of reworking the classification theorem are twofold. First, the theorem is proved in greater generality. Secondly, the commutator relation is never assumed. It is usually replaced by arguments involving lengths of words in the Weyl group. The effect is to clarify the role of the root system in the classification theorem.

Introduction. In this paper we show how to calculate the irreducible characters of the group GL(n, q) of all nonsingular matrices of degree n with coefficients in the finite field of q elements. These characters have been given for n = 2 by H. Jordan [8], Schur [10], and others, and for n =3 and n =4 by Steinberg [12], who has also [13] done important work in the general case. We are concerned here with "ordinary" characters, that is, characters of representations by matrices with complex coefficients. Let Xi, * * *, XA be the distinct absolutely irreducible ordinary characters of a group 5 of order g. By a character of 6 (often called a "generalised character" or "difference character") we mean a class-function 4 on 5 of the form

If there is a nonsingular symmetric bilinear form defined on a distributive algebra A with basis elements C1, C2, … , Cr over a field F such that and , then the so called Casimir operator [1,2]

If k is a field, let G(k) denote the Galois group over k of the algebraic closure of k. In [1 ] Ax proved the following result: Let k be a perfect field such that G(k) is abelian, and every (absolutely irreducible) variety defined over k has a k-rational point. Then G(k) is pro-cyclic. He asked [1, Problem 1] if the assumption that G(k) is abelian can be removed. In this note we construct an example to show that this assumption is necessary. Recall that if X is a variety defined over k, then k(X) is a regular extension of k [2]. Further X contains a k(X)-rational point (in fact a k(X)-rational point which is generic over k). [ADDED IN PROOF. A negative answer to Probelm 1 of [1] has also been given by Moshe Jarden (Rational points on algebraic varieties over large numberfields, Bull. Amer. Math. Soc. 75 (1969), 603-606).] If E and F are regular extensions of k, then the free composite of E and F over k is again a regular extension of k. Let { Ea } be any set of regular extensions of k. By an easy transfinite induction we can construct the free composite of { Ea } over k, and it will again be a regular extension of k. Now let { Ea } be a complete set of nonisomorphic finitely generated regular extensions of k, and let E(k) denote their free composite. Then E(k) is a regular extension of k and every variety X, defined over k, has an E(k)-rational point. Set En(k) =E(En-1(k)) and E??o(k) = Un. 01 E'n(k) . If X is a variety defined over EX(k), then X is actually defined over En(k) for some n, and hence has an En+l(k)-rational point. Since k is algebraically closed in EX(k), the map G(EX(k))-*G(k) is surjective, so G(k) nonabelian implies G(E0(k)) nonabelian. Finally if k is of characteristic zero, then EX(k) is perfect. Hence if we take, say k = Q, we get the desired example. REMARK. In an oral communication, S. Garfunkel has pointed out to Ax that Problem 5 of [1 ] has an affirmative solution, i.e. that the theory of the rings Z/m is decidable. This follows from Corollaries

J. H. M. Wedderburn,l in the course of his fundamental investigations on associative algebras proved that if A is a central2 simple algebra over the field F, there exist extension fields K of F such that the extension AR C A X K of A to the ground field K is isomorphic to a complete matric algebra of a certain degree n over K. The fields K with this property have been called the splitting fields of A. Since A is contained in AK, it follows that A possesses an absolutely irreducible representation by matrices with coefficients in K, and the splitting fields of A can be characterized by this fact. This already shows the close relation between Wedderburn's theory and I. Schur's investigations on the representations of (semi-) groups by linear transformations. This connection was studied later more closely by Emmy Noether and the author.3 It was shown that the splitting fields of finite degree of A can be characterized as the maximal (commutative) subfields of the algebras B of the class of A. Here, two simple algebras A and A1 over F belong to the same class, if in their Wedderburn decompositions A = A X M, Al = Al X M, as direct products of division algebras A, A1 and complete matric algebras M, M, the factors A and A1 are isomorphic. If the Wedderburn factor A of A has the rank m2 over F, m is called the index of the algebra class of A. Then m divides the degree of any splitting field of A, and there exist splitting fields of degree m. When we start now from an arbitrary extension field K of degree n over F, then K may or may not4 be a splitting field of central simple algebras A of a given index m dividing n. In this manner, the theory of algebras provides a classification of extension fields K of a given field. The problem arises to discuss the significance of this classification from the viewpoint of the theory of fields and their Galois groups. That this is possible can be expected because, roughly speaking, the study of simple algebras is equivalent to an investigation of field theoretical questions, as was shown by the author in an earlier paper.' Even when K is a normal field over F, the answer to our question cannot be given in

Introduction. Let G1 and G be connected semisimple algebraic groups defined over a field K of characteristic zero and assume that there is an isomorphism f of G1 onto G which is defined over R, the algebraic closure of K. If p: G -? GL(V) is an absolutely irreducible (finite-dimensional) representation of G defined over K, then p of is an absolutely irreducible representation of G1 defined over K. Satake [7, p. 230] has shown that there is a field K1 which is a finite extension of K, a (unique) central simple division algebra K# defined over K1, a finite-dimensional right vector space V1 over K#, and a K1-homomorphism pi: G1 -GL(V1/K#) (the group of all nonsingular K#-linear endomorphisms of V1) such that (p of)(g) = 01(p1(g)) for all g E G1 where 01 is a unique absolutely irreducible representation of End (V1/K#) (the algebra of all K#-linear endomorphisms of V1) onto End (V). In this paper we are interested in the case where K= K1 and where there are invariant forms on V and V1. More precisely, we state the following two problems. PROBLEM 1. Assume that K#=K and that there are invariant bilinear forms B on V and B1 on V1 which are defined over K. What is the relationship between these two forms over K? Of course, if B is alternating, so is B1 and both are determined by dim V=dim V1. Hence, we shall always take B and B1 to be symmetric. PROBLEM 2. Assume that K# is a nontrivial division algebra over K (i.e., K# AK) and that there is an invariant bilinear form B on V and an invariant ?hermitian form F (c = + 1 or 1) on V1 both of which are defined over K. What is the relationship between these two forms over K? We are especially interested in the case K= Q,, a p-adic field. (In a future paper, we shall discuss the case K=R.) Here, some simplifications are immediately available. In Problem 2, it can be shown [7, p. 232] that K# has an involution of the first kind; but over Q,, it is known that the only such division algebra is the quaternion division algebra. Furthermore, it is known that a hermitian form on a finite-dimensional vector space over a quaternion division algebra defined over Q, is determined only by the dimension of the vector space. Therefore, in Problem 2 we shall always take F to be skew-hermitian; in the case where K# is a quaterion division algebra, this means that the form B is symmetric [7, p. 233]. If W is a vector space defined over K and if S is a symmetric form on W which is also defined over K, then three invariants can be associated with the pair (W, S)

In the arithmetic theory of algebraic varieties it is useful to have a general theory on "reduction modulo 4" of varieties. Here "reduction" means, roughly spoken, the following process: let V be an algebraic variety defined over a field k with a discrete valuation, and denote by p the maximal ideal of that valuation; then we have an algebraic variety VP over the residue field K of t defined by the equations for v modulo p; such a variety V may be called the variety obtained from the variety V by the reduction modulo p. When one tries to construct the general theory of reduction in this sense, the first thing to do will be to investigate how the algebraico-geometric properties of V behave in the reduction process. The object of this paper is concerned with that problem. We shall give a precise definition of the reduction of varieties and the reduction of cycles on a variety. In the case when the ambient varieties v and V are both absolutely irreducible, we shall show that the reduction of cycles on V defines a linear mapping from the group of rational 1-cycles over Ic into the group of rational V-cycles over K of the same dimension; and shall show that the operations on cycles (the intersection-product, the direct product and the algebraic projection) are all preserved by that mapping (Section 4). If we consider reduction as specialization, this result may be regarded as a specialization-theory of cycles on algebraic varieties, in a certain sense. In fact, from this reduction-theory of cycles we can easily obtain a more general specialization-theory of cycles (Section 5). The latter will be a generalization of the specialization-theory of cycles which has been given by T. Matsusaka [4] and P. Samuel [6]. Our theory may be also called a "local" theory of varieties at a divisor p, whereas 4 is not a usual divisor on varieties. We shall also show the following fact: let K be a field with a set of infinitely many valuations which satisfies a certain condition (condition (I) in Section 6); if v is an absolutely irreducible variety defined over K, then for almost all those valuations, it remains absolutely irreducible by the reduction process.

The Brauer-Speiser theorem asserts that the Schur index of a real-valued complex irreducible character of a finite group is either 1 or 2. In this paper we present a brief proof of this result. From this it follows that the K-central nonttrivial division algebra components of group algebras over a real algebraic number field K are quaternions. THEOREM (BRAUER-SPEISER [2], [5]). Let X be a real-valued irreducible complex character of thefinite group G. If X(l) is odd, then the Schur index mQ(X) of X over the raltionals Q equals 1. If X(l) is even, then mQ(X) equtals 1 or 2. PROOF. Let H be the direct product of G with itself and let Go-G be the diagonal of H. We identify G writh Go. Let S be an irreducible complex representation of G affording X and let SW'1 denote the representation of H obtained by taking the outer tensor product of S with itself. S(2) is an irreducible representation of H [4, p. 189]. Denote the clharacter of S(2) by x(2) so that X(2)(gl, g2) =X(g1)X(g2). Thus X2a1 G=X'. Let p be the 1-character of G. By Frobenius reciprocity and the orthogonality relations, (pH, X('))H = (p, X(2) 1 G)a = (p, X')o 1, since x-= Since the representation of H affording p" can be realized in Q, it follows from the theory of the Schur index that mQ(X(2)) 1 [4, ?70]. Let K = Q(X) and let A be the simple component of the group algebra K(G) such that S is an absolutely irreducible constituent of the representation of G afforded by a minimal left ideal of A. Siince K(H) -iK(G)OKcK(G), we see that S(2) is an absolutely irreducible constituent of the representation of H afforded by a minimal left ideal of A(2) =A 0KA. Since mnQ(X(2)) 1, it follows that A(2) is a complete matrix algebra over K [4, ?70]. Tlhus A has expoinent equal to 1 or 2 in the Brauer group of K. Since K is an algebraic number field, the exponent of A equals the index nQ(X) of A. (See [1].) The BrauerSpeiser theorem now follows from the fact that mQ(X) divides X(1) [4, Theorem 70.12]. Received by the editors November 25, 1969. AMS Subject Classifications. Primary 2080; Secondary 1640. Key W1ords and Phrases. Schur index, Brauer group, exponent. I This work was done under the sponsorship of NSF Grant GP-13164/Paige.