| In high school algebraic equations in one unknown of first and second
degree are studied in detail. One learns that for solving these equations
there exist general formulae expressing their roots in terms of the coefficients
by means of arithmetic operations and of radicals. But very
few students know whether similar formulae do exist for solving algebraic
equations of higher order. In fact, such formulae also exist for equations
of the third and fourth degree. We shall illustrate the methods for solving
these equations in the introduction. Nevertheless, if one considers
the generic equation in one unknown of degree higher than four one finds
that it is not solvable by radicals: there exist no formulae expressing the
roots of these equations in terms of their coefficients by means of arithmetic
operations and of radicals. This is exactly the statement of the
Abel theorem.
One of the aims of this book is to make known this theorem. Here we
will not consider in detail the results obtained a bit later by the French
mathematician Évariste Galois. He considered some special algebraic
equation, i.e., having particular numbers as coefficients, and for these
equations found the conditions under which the roots are representable
in terms of the coefficients by means of algebraic equations and radicals.
The book contains many notions which may be new to the reader.
To help him in orienting himself amongst these new notions we put at
the end of the book an alphabetic index of notions, indicating the pages
where their definitions are to be found. |
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 | | Reasoning About Program TransformationsThe motivation of this text lies in what we believe is the inadequacy of current frameworks to reason about the flow of data in imperative programs. This inadequacy clearly shows up when dealing with the individual side effects of loop iterations. Indeed, we face a paradoxical situation where, on the one hand, a typical program spends most of its... | | |
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