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26 CHAPTER 1. INTRODUCTION
Theorem 2 (“Fundamental Theorem of Algebra”) 13 C, the set of complex
numbers, is algebraically closed.
Definition 17 If F is a field, the algebraically closure of F , denoted F is the
field generated by adjoining to F the roots of all polynomials over F .
It follows from proposition 64 that the algebraic closure is in fact algebraically
closed.
1.4 Some Complexity Theory
As is usual in computing we will use the so-called “Landau notation” 14 to describe
the computing time (or space) of operations.
Notation 5 (“Landau”) Let N be some measure of the size of a problem,
generally the size of the input to a program, and f some function. If t(N) is
the time taken, on a given hardware/software configuration, for a particular
program to solve the hardest problem of size N, we say that
if ∃C ∈ R, M : ∀N > Mt(N) < Cf(N).
t(N) = O(f(N)) (1.8)
Hardware tends to change in speed, but generally linearly, so O-notation is independent
of particular hardware choices. If the program implements a particular
algorithm, we will say that the algorithm has this O behaviour.
We will also use “soft O” notation.
Notation 6 (“soft O”) Let N be some measure of the size of a problem, and
f some function. If t(N) is the time taken, on a given hardware/software configuration,
for a particular program to solve the hardest problem of size N, we
say that
t(N) = Õ(f(N)) (1.9)
if t(N) grows “almost no faster” than f(N), i.e. slower than f(N) 1+ɛ for any
positive ɛ: in symbols ∀ε > 0∃C ∈ R, M ∈ N : ∀N > M t(N) < Cf(N) 1+ε .
We should note that “= O” and “= Õ” should really be written with “∈” rather
than “=”, and this use of “=” is not reflexive, symmetric or transitive. Also,
many authors use Õ to mean “up to logarithmic factors”, which is included in
our, more general, definition. [DL08].
The key results of complexity theory for elementary algorithms are that it
is possible to multiply two N-bit integers in time Õ(N), and two degree-N
polynomials with coefficients of bit-length at most τ in time Õ(Nτ). [AHU83]
13 The title is in quotation marks, since C (and R) are constructs of analysis, rather than
algebra.
14 Though apparently first introduced by[Bac94, p. 401]. See [Knu74].