Advanced Abstract Algebra - Maharshi Dayanand University, Rohtak
Advanced Abstract Algebra - Maharshi Dayanand University, Rohtak
Advanced Abstract Algebra - Maharshi Dayanand University, Rohtak
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UNIT-IV 97<br />
If a n = 0 for all n ≥ j and b n = 0 for all n ≥ k , then<br />
c n = 0 for all n ≥ (j+k) .<br />
Thus product of two polynomials is again a polynomial.<br />
The set P of all polynomials over a ring R form a ring under these operations of addition and<br />
multiplication.<br />
Let (a 0 , a 1 , a 2 ,…, a n ,…) , (b 0 , b 1 , b 2 ,…) , (c 0 , c 1 , c 2 ,…) ∈ P.<br />
Then<br />
(i)<br />
(a 0 , a 1 , a 2 ,….) + [(b 0 , b 1 , b 2 ,….) + (c 0 , c 1 , c 2 ,…)]<br />
= (a 0 , a 1 , a 2 ,..) + [(b 0 + c 0 , b 1 +c 1 , b 2 +c 2 ,…)<br />
= (a 0 +b 0 +c 0 , a 1 +b 1 +c 1 , a 2 +b 2 +c 2 ,…)<br />
= (a 0 +b 0 , a 1 +b 1 , a 2 +b 2 ,…) + (c 0 , c 1 , c 2 ,…)<br />
= [(a 0 , a 1 , a 2 ,…) + (b 0 , b 1 , b 2 ,…)] + (c 0 , c 1 , c 2 ,…)<br />
(ii)<br />
(a 0 , a 1 , a 2 ,….) + (0, 0, 0, …) = (a 0 + 0, a 1 + 0, a 2 + 0, …)<br />
= (a 0 , a 1 , a 2 ,….)<br />
and<br />
(0, 0, 0,….) + (a 0 , a 1 , a 2 ,…) = (0 + a 0 , 0 + a 1 , 0 + a 2 ,…)<br />
= (a 0 , a 1 , a 2 ,….)<br />
(iii)<br />
and<br />
(iv)<br />
(v)<br />
where<br />
(a 0 , a 1 , a 2 ,…) + (−a 0 , −a 1 , −a 2 ,…)<br />
= (a 0 − a 0 , a 1 −a 1 , a 2 −a 2 ,…)<br />
= (0, 0, 0, ….)<br />
(−a 0 , −a 1 , −a 2 ,…) + (a 0 , a 1 , a 2 ,….)<br />
= (0, 0, 0,….)<br />
(a 0 , a 1 , a 2 ,…) + (b 0 , b 1 , b 2 ,…) = (a 0 + b 0 , a 1 +b 1 , a 2 +b 2 ,…)<br />
= (b 0 + a 0 , b 1 +a 1 , b 2 +a 2 ,…)<br />
= (b 0 , b 1 , b 2 ,…) + (a 0 , a 1 , a 2 ,…)<br />
[(a 0 , a 1 , a 2 ,…) (b 0 , b 1 , b 2 ,…)] (c 0 , c 1 , c 2 ,…)<br />
= (d 0 , d 1 , d 2 ,…) (c 0 , c 1 , c 2 ,..)<br />
where<br />
d n = a<br />
b j k<br />
j+ k=<br />
n<br />
= (e 0 , e 1 , e 2 ,…)<br />
e m = d<br />
c p q<br />
p+ q=<br />
m<br />
<br />
= <br />
j<br />
p+ q=<br />
m j+<br />
k=<br />
p<br />
<br />
a b<br />
= a jbkcq<br />
.<br />
j+ k+<br />
q=<br />
m<br />
Similarly, it can be shown that<br />
k<br />
<br />
c<br />
<br />
q