An expression of the form polynomial in F of degree n if the coefficientsa ,. . . f a,., are elements of the field F and aQ ^ 0. Multiplication and addition of polynomials are performed in the usual way
A polynomial in F is called reducible in F if it is equal to the product of two polynomials in F each of degree at least one. Polynomials which are not reducible in F are called irreducible in F.
If f (x ) = g(x) . h (x ) is a relation which holds between the polynomials f (x ), g (x ), h (x ) in a field F, then we shall say that g (x ) divides J (x ) in F, or that g ( x ) is a factor of f ( x ). It is readily seen that the degree of f(x) is equal to the sum of the degrees of g (x ) and h (x ), SO that if neither g ( x ) nor h ( x ) is a constant then each has a degree less than f(x). It follows from this that by a finite number of factorizations a polynomial can always be expressed as a product of irreducible polynomials in a field F.
For any two polynomials f (x ) and g (x ) the division algorithm holds, i.e., where q(x) and r(x) are unique polynomials in F and the degree of r (x ) is less than that of g(x). This may be shown by the same argument as the reader met in elementary algebra in the case of the field of real or complex numbers. We also see that r(x) is the uniquely determined polynomial of a degree less than that of g (x ) such that f(x) - r (x ) is divisible by g (x ). We shall call r (x ) the remainder of f (x ).
Also, in the usual way, it may be shown that if a is a root of the polynomial f (x ) in F than x - a is a factor of f (x ), and as a consequence of this that a polynomial in a field cannot have more roots in the field than its degree.
Lemma. If f(x) is an irreducible polynomial of degree n in F, then there do not exist two polynomials each of degree less than n in F whose .product is divisible by f(x).
By the division algorithm, where the degree of r (x ) is less than that of g(x) and r (x ) 1 0 since f(x) was assumed irreducible. Multiplying by h (x ) and transposing, we have
Let us suppose to the contrary that g(x) and h(x) are polynomials of degree less than n whose product is divisible by f(x). Among all polynomials occurring in such pairs we may suppose g(x) has the smallest degree. Then since f(x) is a factor of g(x) . h (x ) there is a polynomial k(x) such that
from which it follows that r(x) . h (x ) is divisible by f (x ). Since f (x ) has a smaller degree than g(x), this last is in contradiction to the choice of g (x )7 from which the lemma follows.
As we saw, many of the theorems of elementary algebra hold in any field F. However, the so-called Fundamental Theorem of Algebra, at least in its customary form, does not hold. It will be replaced by a theorem due to Kronecker which guarantees for a given polynomial in F the existence of an extension field in which the polynomial has a root. We shall also show that, in a given field, a polynomial can not only be factored into irreducible factors, but that this factorization is unique up to a constant factor. The uniqueness depends on the theorem of Kronecker.