The union of and has thus elements. The remaining elements of generate in the sense that no other subfield contains any of them. It follows that they are roots of irreducible polynomials of degree over . This implies that, over , there are exactly irreducible monic polynomials of degree . This may be verified by factoring over .
The union of and has thus elements. The remain.ing elements of generate in the sense that no other subfield contains any of them. It follows that they are roots of i:rreducible polynomials of degree over . This implies that, over , there are exactly irreducible monic polynomials of degree . This may be verified by factoring over .