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Let k, A and B be commutative rings such that A and B are ^-algebras. In this paper it is shown that Ω(kq\A(g)kB), the module of high order differentials of A (x)A B can be expressed by making use of 42? }(A) and ΩkJ)(B). On the other hand let K/k be a finite purely inseparable field extension. Sandra Z. Keith has given a criterion for a /b-linear mapping of K into itself to be a high order derivation of K\k. The representation of Ω(kq)(A®kB) is used to show that Keith's result is valid for larger class of algebras.

LetK be the rational fieldQ or a complex quadratic number field other than\(Q(\sqrt { - 3} )\). LetL be a normal three-dimensional field extension onK. IfR andS are the rings of algebraic integers ofK andL respectively, then the Amitsur cohomology groupH 2 (S/R, U) is trivial. Inflation and class numbers give information about cohomology arising from certain nonnormal cubic extensions.

Let be a finite extension of the field of rational -adic numbers , the ring of integers of , a finite group, the ring of -representations of and ( is the ring of rational integers and the rational number field). We study the algebra in the case where the number of indecomposable -representations of is finite. In particular, for a -group and we find a list of the tensor products of indecomposable -representations of and obtain a description of the radical of and of the quotient algebra . It turns out that in this case we always have .Bibliography: 26 items.

Suppose that C is an algebraically closed field and that Q is a subfield of C. If S is a nonempty subset of C disjoint from Q, it follows from an application of Zorn's lemma that there is a subfield k of C which is maximal with respect to the properties that QCk and k and S are disjoint. The problem is to describe the field extension C/k. When S consists of a single element this has been done by Quigley [4, Theorems 1, 2 and 3]. In this note we shall give several theorems which describe C/k when S consists of exactly two elements. When S contains more than two elements, some of the arguments used in the proof of Theorem 2 fail. The first theorem holds when S is any finite (nonempty) subset of C disjoint from Q. I t generalizes one of Quigley's results [4, Lemma 1].

We define and discuss equations on Banach rings (algebras) which are of polynomial form. We prove a local uniqueness theorem for the homogeneous case, and an existence and local uniqueness theorem for the nonhomogeneous case. In order to apply these results to the equations of Lagrangian quantum field theory we find it necessary to extend the concept of a ring to that of an n‐ring. The resulting theory is applied to a simple model equation arising in quantum field theory.

Let K be a purely inseparable extension of a field k of characteristic p, v’i , i = l)...) m be k-linear maps p’i : K -+ K, and t be a variable. Denoting by 1 the identity map of K, we call C#J~ = 1 + tvl + *a* + tmrpm an “approximate automorphism” of order m of K/k if G,(d) = (@,a)(@,b) mod tm+l for all a, b E K. If for every a E K with a $ k there is some approximate automorphism @, such that Qlta # a (or equivalently such that some yia # 0), then we shall say that K has “enough” approximate automorphisms. This is analogous to the requirement of normality for a finite separable extension, the latter having “enough” genuine automorphisms. It is reasonable to expect, therefore, that a purely inseparable extension with enough approximate automorphisms has further extraordinary properties, and in fact, Sweedler [2] has shown the following

We assume F is a purely inseparable field extension of the field K. The characteristic of K is p ≠ 0, and we assume F and K are not perfect. For x ∈ F, the exponent of x over K is the smallestnon-negative integer e such that and will be denoted bye (x); ⨱ will denote . For any subset S of F, e(x; S) will denote the exponent of x over K(S); in case S = {y} we will write e(x; y) for e(x; S).

Introduction. Let C be a field of characteristic p 0, and let K be a finite algebraic extension field of C such that the pth power of every element of K lies in C. Then K/C is called a purely inseparable extension of exponent 1. It was shown by N. Jacobson that there is a Galois theory for such extensions in which the place of the Galois group is taken by the derivation algebra of K/C. In particular, if K is any field of characteristic p, the purely inseparable extensions K/C of exponent 1 are precisely those in which C is the field of constants of a restricted K-Lie ring of derivations of K which is of finite dimension over K. In the classical theory of simple algebras, it is shown that, if K/C is a Galois extension, the Brauer similarity classes of the simple algebras with center C and split by K constitute a group which is canonically isomorphic with the group of equivalence classes of the group extensions of the multiplicative group of K by the Galois group of K/C. The present paper provides the answer to a question put to me by J-P. Serre, of whether one could establish an analogous result, for K/ C purely inseparable of exponent 1, in which restricted Lie algebra extensions [2] of K by the derivation algebra of K/C take the place of the group extensions. Not only is the answer to this question affirmative, but it provides an excellent illustration of the theory of restricted Lie algebra extensions. It turns out, in fact, that the Lie algebra extensions which arise from simple algebras are trivial extensions when regarded as ordinary extensions, so that the essential structural elements are here precisely those which differentiate the restricted extensions from the ordinary ones. ?1 contains the field theoretical background of our problem. In particular, it gives a simple proof of the main theorem of Jacobson's Galois theory [4] which we include here because it gives us the connection, on which many of our subsequent arguments are based, between the structure of the field extension K/C and that of the derivation algebra of K/C. Theorem 2, which is not needed in the sequel, is the analogue for the present situation of a well known result in the classical Galois theory and is significant for the cohomology theory of derivation algebras. In ?2 we give a proof of a theorem of Jacobson's on derivations (in a slightly generalized form) which is fundamental for the crossed product theory that follows, in the same way as the analogous theorem for isomorphisms is the source of the classical theory of crossed products. In ?3 we discuss the special type of restricted Lie algebra

It is well known that if F is a field of characteristic zero and K= F(cq, ..., ,Cc) is a finite algebraic extension of F, then K contains a primitive element, i.e. an element a such that F(al, . . ., a() = F(x). Moreover, by means of Galois theory, it is possible to characterize those elements of the extension field which are primitive. In the case of finite differentially algebraic extensions the theorem without further restrictions is false. Let Q be the field of rational numbers and 8 the usual derivation, i.e., 8q=O for every q E Q. Let c1, . . ., cn, be algebraically independent complex numbers over Q. If (Q , 8) is the differentially algebraic extension of Q where 8c=0 for every c E Q , then the underlying set of Q is identical with that of Q(c1, . . ., cs), whence it is clear that there is no element c E Q such that Q = Q . Kolchin [2] (also [5, p. 52]) has shown the existence of primitive elements in the case where the differential field F has one derivation operator and the field F has an element f such that 8f# 0(1). The differential fields (F , D) considered in this paper are differentially algebraic over F, but F does not contain nonconstant elements. We prove the existence of primitive elements in the case where the derivation operator satisfies the conditions

Classical Galois cohomological results for purely inseparable field extensions of exponent one are generalized here to commutative rings of prime characteristic. Given a commutative ring extension C over A of prime characteristic p, there are three variants for the Brauer group B(C/A) of central separable A-algebras split by C: the Amitsur cohomology group H2(C/A, Gm), the Chase-Rosenberg group PV(C/A), and Hochschild's group 4(C, g) of regular restricted Lie algebra extensions of C by the Lie algebra g of all A-derivations on C. In this paper we show that if C is finitely generated projective as an A-module and C [g] = EndA (C), then H2(C/A, Gm) , C) ,l(C/A). As a corollary we show that Hi(C/A, Gm) is zero for all i > 2. When C is a field, these are the results of Berkson, Hochschild and Rosenberg and Zelinsky [4], [11], [12]. As in [11] we show that the Lie algebra extensions which arise from central separable algebras are trivial extensions when regarded as ordinary extensions so that the essential structural elements are here precisely those which differentiate the restricted extensions from the ordinary ones. We also show that if R is a commutative C-algebra which is finitely generated, projective as a C-module, then the Brauer group B(R/A) is mapped onto the Brauer group B(R/C). The last result is also due to Hochschild when C is a field [10]. ?1 contains the background on projective modules which came into the picture. Due to their peculiar behavior all relevant automorphisms turn out to be inner which explains why instead of some exact sequences we get two isomorphism theorems. In ?2 the isomorphism of e(g, C) with 91(C/A) is proved. ?3 and ?4 provide the preliminary materials for ?5. ?3 contains an exposition on the theory of differentials in rings of prime characteristic. Its application to Amitsur cohomology is given in ?4. The main results are given in ?5. Throughout this paper C over A always denotes a commutative ring extension of prime characteristic p such that C is finitely generated projective as an A-module Received by the editors October 30, 1969. AMS 1970 subject classifications. Primary 13A20.

Let A be a principal ideal domain, K be the quotient field of A, and let L be a cubic extension of K. In this paper we establish the existence of a special type of integral basis of the field L over K which is a generalization of the integral basis of Voronoi for cubic extensions of the field Q of rational numbers.

An inversive difference field K \mathcal {K} is a field K together with a finite number of automorphisms of K. This paper studies inversive extensions of inversive difference fields whose underlying field extensions are separable algebraic. The principal tool in our investigations is a Galois theory, first developed by A. E. Babbitt, Jr. for finite dimensional extensions of ordinary difference fields and extended in this work to partial difference field extensions whose underlying field extensions are infinite dimensional Galois. It is shown that if L \mathcal {L} is a finitely generated separable algebraic inversive extension of an inversive partial difference field K \mathcal {K} and the automorphisms of K \mathcal {K} commute on the underlying field of K \mathcal {K} then every inversive subextension of L / K \mathcal {L}/\mathcal {K} is finitely generated. For ordinary difference fields the paper makes a study of the structure of benign extensions, the group of difference automorphisms of a difference field extension, and two types of extensions which play a significant role in the study of difference algebra: monadic extensions (difference field extensions L / K \mathcal {L}/\mathcal {K} having at most one difference isomorphism into any extension of K \mathcal {K} ) and incompatible extensions (extensions L / K , M / K \mathcal {L}/\mathcal {K},\mathcal {M}/\mathcal {K} having no difference field compositum).

In w 1 we consider the situation: L/K is a finite separable field extension, A is an abelian variety over L, and A, is the abelian variety over K obtained from A by restriction of scalars. We study the arithmetic properties of A, relative to those of A, and in particular show that the conjectures of Birch and Swinnerton-Dyer hold for A if and only if they hold for A, . In w 2 we study certain twisted products of abelian varieties and use our results to show that the conjectures of Birch and Swinnerton-Dyer are true for a large class of twisted constant elliptic curves over function fields. In w we develop a method of handling abelian varieties over a number field K which are of CM-type but which do not have all their complex multiplications defined over K. In particular we compute under quite general conditions the conductors and zeta functions of such abelian varieties and so verify Serre's conjecture [12] on the form of the functional equation. Similar, but less complete, results have been obtained by Deuring [1] for elliptic curves and Shimura [15] for abelian varieties.

in B. The classical method of field extension (forming the polynomial ring A [x] and reducing modulo the principal ideal J of 1.1) solves this problem. In this paper, we investigate the role of this process in topological rings. We study the problem of obtaining a normed linear algebra extension B containing a solution of 1.1, where A is a given normed linear algebra. Our method is to define a norm first in the polynomial algebra A [x], and to do it in such a way that J is closed (for otherwise the canonical norm in

Let k be a finite extension of the field of 2-adic numbers. In this paper we determine the structure of the Galois groups of the algebraic closure and of the maximal extension without simple ramification of the field k under the assumption that the maximal extension without higher ramification of the field k contains a fourth root of 1.

The last comparison is that of H*(H, A) with the Amitsur cohomology of A. First we show that there always is a natural transformation from the Amitsur cohomology of A to H*(H, A). We then specialize to the case that A is a finite field extension and give conditions on A and H which imply the natural transformation is an isomorphism. We also show that many field extensions A can

We show that a valued field A with only a finite number of nonisomorphic valued extensions is equal to the complex field C or is real closed with C=A(x/(-1)). The Ostrowski (Gelfand-Mazur) Theorem [2, p. 131], [4, p. 260] implies that with any of the valuations v(x) = I xj t, where j * I denotes the usual modulus and 0

Throughout this note L/K denotes a purely inseparable field extension of characteristic p and nonzero exponent. In [5, p. 745], Rygg proves that when L/K has bounded exponent, then a subset M of L is a relative ?-base of L/K if and only if M is a minimal generating set of L/K. The purpose of this note is to answer the following question : If every relative ?-base of L/K is a minimal generating set, then must L/K he of bounded exponent? The answer is known to be yes when K and 2>* are linearly disjoint, * = 1, 2, • • • , see [l]. We give two examples for which the answer is no: One in which the maximal perfect subfield of L is contained in K, and the other in which it is not. The following lemmas are needed for our examples. An intermediate field V of L/K is called proper if KCLL'EL.

Van der Waerden (1930 a , pp. 128- 131) has discussed the problem of carrying out certain field theoretical procedures effectively, i.e. in a finite number of steps. He defined an ‘explicitly given’ field as one whose elements are uniquely represented by distinguishable symbols with which one can perform the operations of addition, multiplication, subtraction and division in a finite number of steps. He pointed out that if a field K is explicitly given then any finite extension K' of K can be explicitly given, and that if there is a splitting algorithm for K , i.e. an effective procedure for splitting polynomials with coefficients in K into their irreducible factors in K [x], then(1) there is a splitting algorithm for K' . He observed in (1930 b ), however, that there was no general splitting algorithm applicable to all explicitly given fields K , or at least that such an algorithm would lead to a general procedure for deciding problems of the type ‘Does there exist an n such that E(n) ?’, where E is an arbitrarily given property of positive integers such that there is an algorithm for deciding for any n whether E(n) holds. In this paper we review these results in the light of the precise definition of algorithm (finite procedure) given by Church (1936), Kleene (1936) and Turing (1937) and discuss the existence of a number of field theoretical algorithms in explicit fields, and the effective construction of field extensions. We sharpen van der Waerden’s result on the non-existence of a general splitting algorithm by constructing (§7) a particular explicitly given field which has no splitting algorithm. We show (§7) that the result on the existence of a splitting algorithm for a finite extension field does not hold for inseparable extensions, i.e. we construct a particular explicitly given field K and an explicitly given inseparable algebraic extension K ( x ) such that K has a splitting algorithm but K ( x ) has not. (2) We note also (in §6) that there exist two isomorphic explicitly given fields, one of which possesses a splitting algorithm but the other of which does not. Thus the sort of properties of fields we are interested in depend not only on the abstract field but also on the particular representation chosen. It is necessary therefore to state rather carefully our definitions of explicit ring, extension ring, splitting algorithm, etc., and to introduce the concept of explicit isomorphism (3) and homomorphism. This occupies §§ 1,2 and 3. On the basis of these definitions we then discuss the existence of some fundamental field theoretical algorithms in explicit fields and their extension fields. This leads also to a classification of the types of extension fields which can be effectively constructed.

1. Let F be an extension field of K. A minimal set of generators of F over K is a subset S of F such that F=K(S) and S'CS implies K(S') CK(S) where C denotes proper inclusion. Pickert [4, p. 881 has shown that if F is a finite inseparable extension of K (the characteristic of K is p O) and S= {a,, * * *, an} is a minimal set of generators of F over K, then S is p-independent in F (this concept, due to Teichniiller [5], is defined in ?2 following) and is a minimal set of generators of F over FP(K). A relative p-basis of F over K, as introduced in [5], is a minimal set of generators of F over FP(K). It is shown by Becker and MacLane [i, Theorem 6] that if F is a finite purely inseparable extension of K, then the minimal number of generators of F over K is n, the exponent determined by the degree [F: FP(K) ] = pn. Closely related results are given by Weil [6, Chapter I, ?5] and by Zariski and Samuel [7, Chapter II, ?171 in a discussion of derivations on fields. In this note we assume that F is a purely inseparable extension of K of arbitrary degree but with finite exponent e: FP CK. It is the purpose of this note to prove the following

The purpose of this paper is to obtain quantitative results on invertible powers of (fractional) ideals in commutative separable algebras over dedekind domains. This is connected with the work of Dade, Taussky and Zassenhaus(2) on ideals in noetherian domains. We do not, however, make use of their paper, but rather draw on the general theorems on ideals in commutative separable algebras established by Fröhlich (3), in particular his qualitative result that some power of any given ideal is invertible. Our basic result (Theorem 1) concerns the invertibility of powers of a particular type of ideal, the componentwise dedekind ideals defined below. From this we deduce a general result (Theorem 2), which includes as a special case Theorem C of (2) for the case of separable field extensions; specifically, the (n – 1)th power of any ideal is invertible, where n is the dimension of the algebra. Although, as we show, it is possible to deduce Theorem 2 from (2), we have here an independent proof of one of the main results of (2) based entirely on the results in (3). As a further application of Theorem 1 we obtain a new result on ideals over the group ring of an abelian group over the ring of rational integers; the (t – 1)th power of such an ideal is invertible, where t is the maximum number of simple components of the rational group algebra of any Sylow subgroup. We also show that this is the best possible result when some Sylow subgroup whose rational group algebra has t components is cyclic.

and Fk* the group of units of Fk. Define ring homomorphisms ei (i = 1, , k+ 1) of Fk to Fk+l by(2'3) ei(fl? * ?fk) =fl? * fi?1 fi ... ?fk and define a multiplicative homomorphism Ak: Fk* -> F(k+l)* by Ak(X) = el(x) (E2(X)) * ... (ek+l(x))+1. Then the groups Fk* and mappings Ak form the Amitsur complex; the kth homology group Ker Ak+l/Im Ak we denote by Hk(F). In case F is a finite dimensional extension field, Amitsur showed that H2(F) is isomorphic to the Brauer group of central simple C-algebras split by F. He also showed that in case F is a normal separable extension, Hn(F) is isomorphic to Hn(G, F*) the nth cohomology group of the Galois group G of F over C with coefficients in the group of nonzero elements of F. In this paper, we extend and simplify Amitsur's results. We begin by showing (?2) that in case F is a separable field extension of C, Hn(F) --'Hn([G: H], K*), where K is a normal closure of F with Galois group G, H is the subgroup corresponding to F, and the cohomology group on the right side is the relative cohomology group as introduced in [I]. Next, we study Hn(F) when C is not necessarily a field but when n = 2. In ?3, under weak hypotheses on F, we exhibit a homomorphism of H2(F) to the (generalized) Brauer group of central separable algebra classes split by F [5]. This homomorphism is an isomorphism under stronger hypotheses on F and C. These hypotheses are slightly weaker than assuming all projective C, F, and F2 modules are free and include the cases (10) C is semilocal (not necessarily Noetherian) and F is a C-algebra which is a-finitely generated projective C-module containing C 1 as a direct summand and (20) C=K[x], F=L[x], with K a field and L a finite dimensional commutative K-algebra. Hochschild in [13] has given a description of the Brauer group in case F is a purely inseparable extension field of C of exponent 1. In [3, ?7], Amitsur

Let K K be a field having prime characteristic p p . The following conditions on a subfield k k of K K are equivalent: (i) ∩ n K p n ( k ) = k { \cap _n}{K^{{p^n}}}(k) = k and K / k K/k is separable. (ii) k k is the field of constants of an infinite higher derivation defined in K K . (iii) k k is the field of constants of a set of infinite higher derivations defined in K K . If K / k K/k is separably generated and k k is algebraically closed in K K , then k k is the field of constants of an infinite higher derivation in K K . If K / k K/k is finitely generated then k k is the field of constants of an infinite higher derivation in K K if and only if K / k K/k is regular.

Introduction. Let the difference field2 j have the transformally algebraic extension3 5. We have previously defined4 two numerical invariants, order and effective order, of such an extension. In this note we introduce a new numerical invariant, the limit degree of the extension 5 of j, which will be denoted by l.d. (5/W). We suppose, first, that 5 is finitely generated; say 5 = j(ai,* .r). Let Sk denote the set of the atj and their first k transforms. We denote by dk the degree of F(Sk+1) with respect to F(Sk), k = 0, 1, 2, * . . The dk form a nonincreasing sequence whose terms are finite integers for sufficiently large k. We define l.d. (5/j) to be the least value assumed by the dk. It will be proved in ?1 that the limit degree is independent of the choice of the set of generators ai. If 5 is any-not necessarily finitely-generated-transformally algebraic extension of j, we define l.d. (5/j) to be the maximum of the limit degrees of all finitely-generated sub-extensions of j, if this maximum exists, or oo, if it does not. That this definition is consistent with the preceding follows from Theorem I below when it is restricted to finitely generated extensions. If 9) is an irreducible manifold of dimension zero over a difference field j and 2 the corresponding prime difference ideal, we define the limit degree of 9) over j, or of 2, to be l.d. (5/ ), where 5 is the field obtained by adjoining a generic zero of 2 to j. If 5 is not transformally algebraic over F the definition of limit degree may be applied and will always result in the value oo. In this case it is more useful to consider the limit degree relative to a given basis of transformal transcendency. If S is the field formed by adjoining this basis to a, then l.d. (5/a) relative to the given basis is

Since Riemann's dissertation of 1851, Riemann surfaces have for the most part been considered as suitable domains of definition for analytic functions. Here, however, we view them as topological spaces associated with certain kinds of field extension, and consider how their topological properties are connected with algebraic properties of these field extensions. A specialization of these results gives us arithmetic properties of fields of algebraic functions of one real variable.

We give a new proof of the theorem that Amitsur’s complex for purely inseparable field extensions has vanishing homology in dimensions higher than 2. This is accomplished by computing the kernel and cokernel of the logarithmic derivativet →Dt/t mapping the multiplicative Amitsur complex to the acyclic additive one (D is a derivation of the extension field).

We define a diagram associated with a purely inseparable field extension of finite exponent. We show that, under this definition, for any given field extension the shape of its diagram is unique. Thus our diagram improves the diagram defined by Sweedler in [2, p. 402]. In §2 we define an “allowable” shape for a diagram. Given any “allowable” shape for a diagram representing a finite field extension, we construct a field extension whose diagram has that shape.

It is a classical result that if ( R , M ) (R,\mathfrak {M}) is a complete discrete valuation ring with quotient field K K , and if R / M R/\mathfrak {M} is perfect, then any finite dimensional central simple K K -algebra Σ \Sigma can be split by a field L L which is an unramified extension of K K . Here we prove that if ( R , M ) (R,\mathfrak {M}) is any regular local ring, and if Σ \Sigma contains an R R -order Λ \Lambda whose global dimension is finite and such that Λ / Rad ⁡ Λ \Lambda /\operatorname {Rad} \Lambda is central simple over R / M R/\mathfrak {M} , then the existence of an “ R R -unramified” splitting field L L for Σ \Sigma implies that Λ \Lambda is R R -separable. Using this theorem we construct an example which shows that if R R is a regular local ring of dimension greater than one, and if its characteristic is not 2, then there is a central division algebra over K K which has no R R -unramified splitting field.

The results obtained here must have been known and settled centuries ago. However, they have proved impossible to locate in the available literature. H. K. Farahat has asked for proofs of the linear independence over the rationals of certain infinite sequences of real numbers such as √2, √3, √5.... He also raised the general question of determining the degree of the field extension generated over the rationals by a family of positive irrational numbers of the form x=a1/mwhere a, m are positive integers.

Necessary and sufficient conditions are given for two K-division rings, K an algebraic number field, to have precisely the same set of subfields. Using this, an example is presented of two K-division rings having precisely the same set of subfields such that only one of the division rings can be embedded in a Q-division ring. Let K be a field. By a K-division ring we mean a finite-dimensional division algebra with center K. If D is a K-division ring and k is a field, k'c K, we say that D is k-adequate if D can be embedded in a k-division ring. Similarly, if L is a field, we say that L is k-adequate if L is a subfield of some k-division ring. Clearly, if D is k-adequate then so is every subfield of D. In [4] the converse was raised: if every subfield of D is k-adequate, must D be k-adequate? We show that the answer to this question is no by exhibiting two K-division rings D1 and D2 having precisely the same set of subfields and such that D1 is k-adequate (and so every subfield of D2 is also k-adequate) but D2 is not k-adequate. Throughout this paper K will denote an algebraic number field. We will use freely the classification theory of K-division algebras by means of Hasse invariants. The reader is referred to [3] for the relevant theory. If a is a prime of K and D is a K-division ring, we denote the Hasse invariant of D at Y by inv,, D. The order of inv. D in Q/Z is denoted by l.i. D. Here Q denotes the field of rational numbers and Z is the ring of ordinary integers. We denote the completion of K at the prime 9 by K,. The dimension of D over K is denoted by [D: K]; we use the same notation for the dimension of field extensions. We begin by establishing criteria for two K-division rings to have precisely the same set of subfields. THEOREM 1. Let D1 and D2 be K-division rings. Then D1 and D2 have precisely the same set of subfields if and only if l.i.g Dj=l.i.,+ D2 for all primes 9 of K. Received by the editors July 19, 1971. AMS 1970 subject classifications. Primary 16A40; Secondary 12A65.

In this chapter we shall analyze arbitrary extension fields of a given field. Since algebraic extensions were studied in some detail in Chapter V, the emphasis here will be on transcendental extensions. As the first step in this analysis, we shall show that every field extension K ⊂ F is in fact a two-step extension K ⊂ E ⊂ F, with F algebraic over E and E purely transcendental over K (Section 1). The basic concept used here is that of a transcendence base, whose cardinality (called the transcendence degree) turns out to be an invariant of the extension of K by F (Section 1). The notion of separability is extended to (possibly) nonalgebraic extensions in Section 2 and separable extensions are characterized in several ways.