In any variety of algebras such as groups or rings the members can be defined by presentations in terms of generators and defining relations. Fields do not form a variety, as we saw in 1.2, and they do not have presentations in this sense. In fact the usual mode of construction of commutative fields is quite different: one takes a rational function field over some ground field and then makes an algebraic extension. For skew fields this method works only in very special cases, but there is an analogue of a presentation, in terms of matrices that become singular. Now it is necessary to verify in each case that the outcome is a field and this forms the subject of 6.1. At this stage free fields form a natural topic, but first we need to prepare some tools: In 6.2 we prove the specialization lemma, which generalizes the density principle of the commutative theory: A non-zero polynomial over an infinite (commutative) field assumes non-zero values. The analogue for skew fields states that a non-zero element of a free field (with infinite centre and of infinite degree over it) can be specialized to a non-zero element in the field.

The elements of the free field are described in terms of matrices over the tensor ring and it is therefore of interest to provide a normal form for these matrices. This is done in 6.3 and a corresponding ‘normal form’ for fractions is derived.