Galois Theory, 4th Edition

52 Factorisation of Polynomials

(3) However, t

2

−2 is reducible over the larger subfield R, for now

t

2

−2 = (t −

√

2)(t +

√

2)

This shows that an irreducible polynomial may become reducible over a larger sub-

field of C.

(4) The polynomial 6t + 3 is irreducible in Z[t]. Although it has factors

6t + 3 = 3(2t + 1)

the degree of 2t + 1 is the same as that of 6t + 6. So this factorisation does not count.

(5) The constant polynomial 6 is irreducible in Z[t]. Again, 6 = 2 ·3 does not count.

Any reducible polynomial can be written as the product of two polynomials of

smaller degree. If either of these is reducible it too can be split up into factors of

smaller degree ...