Let π: G → GL(V, F) be an FG-representation, E a subfield of F, and K an extension field of F. Then V is also a vector space over E with GL(V, F) ≤ GL(V, E), so π also defines an EG-representation. Further, by a tensoring process discussed in section 25, π induces a KG-representation πK on a K-space VK. This chapter investigates the relationship among these representations. It will often be very useful to extend F to K by passing from π to πK. For example several results at the end of chapter 9 are established in this way.

π is said to be absolutely irreducible if πK is irreducible for each extension K of F, and F is said to be a splitting field for G if every irreducible FG-representation is absolutely irreducible. It develops in section 25 that it is absolutely irreducible precisely when F = EndFG(V) and in section 27 that if G is finite then a splitting field is obtained by adjoining a suitable root of unity to F. It's particularly nice to work over a splitting field. For example in section 27 it is shown that, over a splitting field, the irreducible representations of the direct product of groups are just the tensor products of irreducible representations of the factors.

Section 26 investigates representations over finite fields, where change of field goes very smoothly. Lemma 26.6 summarizes many of the relationships involved.