12 Selected References 79Field theory was established as an independent subject of study in 1910 by
Steinitz [68]. The books of Jacobson [41] and Lang [48] treat also the more recent
theory of ordered fields, due to Artin and Schreier (1927).
Fields and groups are connected with one another byGalois theory. This subject
has its origin in attempts to solve polynomial equations ‘by radicals’. The founder of
the subject is really Lagrange (1770/1). By developing his ideas, Ruffini (1799) and
Abel (1826) showed that polynomial equations of degree greater than 4 cannot, in gen-
eral, be solved by radicals. Abel (1829) later showed that polynomial equationscanbe
solved by radicals if their ‘Galois group’ is commutative. In honour of this result,
commutative groups are often calledabelian.
Galois (1831, published posthumously in 1846) introduced the concept of normal
subgroup and stated a necessary and sufficient condition for a polynomial equation to
be solvable by radicals. The significance of Galois theory today lies not in this result,
despite its historical importance, but in the much broader ‘fundamental theorem of
Galois theory’. In the form given it by Dedekind (1894) and Artin (1944), this estab-
lishes a correspondence between extension fields and groups of automorphisms, and
provides a framework for the solution of a number of algebraic problems.
Morandi [54] and Rotman [61] give modern accounts of Galois theory. The histor-
ical development is traced in Kiernan [43]. In recent years attention has focussed on
the problem of determining which finite groups occur as Galois groups over a given
field; for an introductory account, see Matzat [51].
Some texts on linear algebra and matrix theory are Halmos [31], Horn and
Johnson [37], Mal’cev [50] and Gantmacher [28].
The older literature on associative algebras is surveyed in Cartan [13]. The texts on
noncommutative rings cited above give modern introductions.
A vast number of characterizations of inner product spaces, in addition to the par-
allelogram law, is given in Amir [3]. The theory of Hilbert space is treated in the books
of Riesz and Sz.-Nagy [58] and Akhiezer and Glazman [2]. For its roots in the theory
of integral equations, see Hellinger and Toeplitz [33]. Almost periodic functions are
discussed from different points of view in Bohr [9], Corduneanu [17] and Maak [49].
The convergence of Fourier series is treated in Zygmund [73], for example.
12 SelectedReferences
[1] L.V. Ahlfors,Complex analysis, 3rd ed., McGraw-Hill, New York, 1978.
[2] N.I. Akhiezer and I.M. Glazman,Theory of linear operators in Hilbert space, English
transl. by E.R. Dawson based on 3rd Russian ed., Pitman, London, 1981.
[3] D. Amir,Characterizations of inner product spaces,Birkh ̈auser, Basel, 1986.
[4] M.F. Atiyah and I.G. Macdonald,Introduction to commutative algebra, Addison-Wesley,
Reading, Mass., 1969.
[5] F.V. Atkinson, The reversibility of a differentiable mapping,Canad. Math. Bull. 4 (1961),
161–181.
[6] G. Birkhoff,Lattice theory, corrected reprint of 3rd ed., American Mathematical Society,
Providence, R.I., 1979.
[7] G. Birkhoff and S. MacLane,A survey of modern algebra, 3rd ed., Macmillan, New York,
1965.
[8] F. van der Blij, History of the octaves,Simon Stevin 34 (1961), 106–125.