A complete theory of the degree of a polynomial ideal is presented, with a systematic use of the rational form of the Hilbert function in place of the (more commonly used) Hilbert polynomial. This is used for a simple algebraic proof of classical Bézout theorem, and for proving a "strong Bézout inequality", which has as corollaries all previously known Bézout inequalities , and is much sharper than all of them in the case of a non-equidimensional ideal.