Abstract Algebra Theory and Applications - Computer Science ...

15.1 POLYNOMIAL RINGS 259ThenTo show that polynomial multiplication is associative, letm∑p(x) = a i x i ,[p(x)q(x)]r(x) =======q(x) =r(x) =i=0n∑b i x i ,i=0p∑c i x i .i=0[(∑ m) ( n∑)] ( p∑)a i x i b i x i c i x ii=0 i=0i=0⎡ ⎛ ⎞ ⎤ )m+n∑ i∑⎣ ⎝ a j b i−j⎠ x i ⎦ c i x ii=0m+n+p∑i=0m+n+p∑i=0m+n+p∑i=0⎡⎣j=0i∑( j∑j=0 k=0⎛⎝∑⎡⎣j+k+l=ii∑j=0(∑ m) ⎡a i x i ⎣i=0( m∑a k b j−k)( p∑i=0a j b k c r⎞⎠ x ic j⎤⎦ x i( i−j) ⎤ ∑a j b k c i−j−k⎦ x ik=0⎛ ⎞ ⎤n+p∑ i∑⎝ b j c i−j⎠ x i ⎦i=0) [( n∑a i x ii=0 i=0= p(x)[q(x)r(x)]j=0b i x i ) ( p∑i=0c i x i )]The commutativity and distribution properties of polynomial multiplicationare proved in a similar manner. We shall leave the proofs of these propertiesas an exercise.□Proposition 15.2 Let p(x) and q(x) be polynomials in R[x], where R is anintegral domain. Then deg p(x) + deg q(x) = deg(p(x)q(x)). Furthermore,R[x] is an integral domain.