#### 7.3.4 The Lindemannn Weierstrass Theorem And Vector Spaces

As another application of the abstract concept of vector spaces, there is an amazing theorem due
to Weierstrass and Lindemannn.

Theorem 7.3.34 Suppose a_{1},

,a_{n} are algebraic numbers, roots of a polynomial with rational
coefficients, and suppose α_{1},,α_{n} are distinct algebraic numbers. Then
In other words, the

are independent as vectors with field of scalars equal to the
algebraic numbers.
There is a proof of this in the appendix. It is long and hard but only depends on elementary
considerations other than some algebra involving symmetric polynomials. See Theorem
F.3.5.

A number is transcendental, as opposed to algebraic, if it is not a root of a polynomial which
has integer (rational) coefficients. Most numbers are this way but it is hard to verify that specific
numbers are transcendental. That π is transcendental follows from

By the above theorem, this could not happen if π were algebraic because then iπ
would also be algebraic. Recall these algebraic numbers form a field and i is clearly
algebraic, being a root of x^{2} + 1. This fact about π was first proved by Lindemannn in
1882 and then the general theorem above was proved by Weierstrass in 1885. This
fact that π is transcendental solved an old problem called squaring the circle which
was to construct a square with the same area as a circle using a straight edge and
compass. It can be shown that the fact π is transcendental implies this problem is
impossible.