Abstract Algebra and Algebraic Number Theory

Definition 2.2.11 (Polynomial ). Let R be a ring, then an ordered subset
of (a 1 , a 2 , ...., a n , ....) of R is called a polynomial over R if ∃n ∈ N ∪ {0} such
that a n ≠ 0 and a i = 0∀i > n. n is called the degree of the polynomial
and a n is called leading coefficient of the polynomial.
Two polynomials (a 1 , a 2 , ...., a n , ....) and (b 1 , b 2 , ...., b m , ....) are called equal
iff m = n ∧ a i = b i ∀i ∈ N ∪ {0}.
The polynomial (0, 0, ...., 0, ....), in which each coordinate is zero is called
a zero polynomial over R. In practice degree of zero polynomial is
taken to be inf.
Representation of a polynomial: The polynomial (a 1 , a 2 , ...., a n , ....)
with leading coefficient a n is represented by a 0 x 0 + a 1 x + a 2 x 2 + .... + a n x n ,
where x 0 , x 1 , x 2 , ...., x n represents the coordinates of a 0 , a 1 , a 2 , ...., a n respectively
and called indeterminate, having the properties-
• ax + bx = (a + b) x
• x r x s = x r+s = x s+r
• x 0 behaves as a.x 0 = a
Definition 2.2.12 (Addition and Multiplication of polynomials). Let
p (x) = a 0 + a 1 x + a 2 x 2 + .... + a n x n =
q (x) = b 0 + b 1 x + b 2 x 2 + .... + b m x m =
n∑
a i x i
i=0
m∑
b j x j
be polynomials over ring R. Define addition and multiplication of polynomials
as
.
p (x) + q (x) =
p (x) .q (x) =
(m+n)
∑
k=0
max(m+n)
∑
k=0
Theorem 2.2.17. Let R be a ring then
j=0
(a k + b k ) x k
c k x k , where c k = ∑
i+j=k
a i b j
R [x] = {p (x) : p (x) is a polynomial over R}
forms a ring with respect to the addition and multiplication of a polynomials.
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