  The study of Mutually Unbiased Bases continues to be developed vigorously,
and presents several challenges in the Quantum Information Theory. Two
orthonormal bases in $\mathbb C^d, B {and} B'$ are said mutually unbiased if
$\forall b\in B, b'\in B'$ the scalar product $b\cdot b'$ has modulus
$d^{-1/2}$. In particular this property has been introduced in order to allow
an optimization of the measurement-driven quantum evolution process of any
state $\psi \in \mathbb C^d$ when measured in the mutually unbiased bases
$B\_{j} {of} \mathbb C^d$. At present it is an open problem to find the maximal
umber of mutually Unbiased Bases when $d$ is not a power of a prime number.
\noindent In this article, we revisit the problem of finding Mutually Unbiased
Bases (MUB's) in any dimension $d$. The method is very elementary, using the
simple unitary matrices introduced by Schwinger in 1960, together with their
diagonalizations. The Vandermonde matrix based on the $d$-th roots of unity
plays a major role. This allows us to show the existence of a set of 3 MUB's in
any dimension, to give conditions for existence of more than 3 MUB's for $d$
even or odd number, and to recover the known result of existence of $d+1$ MUB's
for $d$ a prime number. Furthermore the construction of these MUB's is very
explicit. As a by-product, we recover results about Gauss Sums, known in number
theory, but which have apparently not been previously derived from MUB
properties.
