Let N denote a normal closure of L/K and let Gal(N/K) denote the

Galois group of N/K.

n] is the Galois group of some subspace in the k-equal arrangement.

1], so all of the subspaces in the k-equal arrangement have irreducible Galois groups.

On this basis rests the subsequent discussion of polynomials and their

Galois groups and representations of the

Galois group, followed by the reciprocity laws, including the famous proof by Andrew Wiles of Fermat's Last Theorem.

With concrete examples he begins with cubic equations, turning to complex numbers, biquadratic equations, equations of degree n and their properties, including plausibility and proof, the search for additional solution formulas, equations that can be reduced in degree, including the decomposition of integer polynomials and Eisenstein's irreducibilty criterion, the construction of regular polygons, the

Galois group of an equation, and algebraic structures and Galois theory, including groups and fields, the fundamental theorem, and Artin's version of the fundamental theory.

He proceeds explaining fields, including extension fields, normal and separable extensions, the

Galois group, and Galois correspondence.

The 23 papers discuss such topics as the embedded eigenvalue problem for classical groups, orbital integrals and distributions, Arthur's asymptotic inner product formula of truncated Eisenstein series, parametrization of tame super-cuspidal representations, icosahedral fibers of the symmetric cube and algebraicity, and motivic

Galois groups and L-groups.

She continues with action-angle variables, integrability and

Galois groups (including the Arnold-Liouville theorem, as revisited through differentia Galois theory), Lax closing with an introduction to Lax equations as they apply to the Arnold-Liouville theorem.

For example, one may well argue that "five" (the abstract quantity, fiveness) exists apart from us--even if humans had never evolved, you could still have five rocks in a field--and perhaps this extends to fractions and even irrational numbers; but when you start to talk about negative numbers, complex numbers, hypercomplex numbers, infinitesimals, transfinites, matrices, vectors, multi-variable functions, tensors, fields,

Galois groups, and the Mandelbrot set the compulsion to regard these as purely mental constructs is overwhelming.

This property is also valid on Galois groups, though applied restrictively.

It is important to notice that, in schemes based on Galois groups.

With examples, exercises and open problems he covers examples low degree, nilpotent and solvable groups as

Galois groups over Q, Hilbert's irreducibility theorem, Galois extensions of Q(T), Galois extensions of Q(T) given by torsion on elliptic curves, Galois extensions of C(T), rigidity and rationality on finite groups, construction of Galois extensions of Q(T) by the rigidity methods, the quadratic form and its applications, and in an appendix, the large sieve inequality.