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Computer Algebra System (CAS) 14-63
Example
Find the solutions P(X) of:
P(X) = X (mod X
2
+ 1)
P(X) = X – 1 (mod X
2
– 1)
Typing:
CHINREM((X) AND (X
2
+ 1), (X – 1) AND (X
2
– 1))
gives:
That is:
CYCLOTOMIC Returns the cyclotomic polynomial of order n. This is a
polynomial having the nth primitive roots of unity as
zeros.
CYCLOTOMIC has an integer n as its argument.
Example 1
When n = 4 the fourth roots of unity are {1, i, –1, –i}.
Among them, the primitive roots are: {i, –i}. Therefore, the
cyclotomic polynomial of order 4 is (X – i).(X + i) = X
2
+ 1.
Example 2
Typing:
CYCLOTOMIC(20)
gives:
EXP2HYP EXP2HYP has an expression enclosing exponentials as an
argument. It transforms that expression with the relation:
exp(a) = sinh(a) + cosh(a).
x
2
2x–1+
2
--------------------------
–
AND
x
4
1–
2
--------------
P
X[]
x
2
2x–1+
2
--------------------------
mod
x
4
1–
2
--------------
–
⎝
⎠
⎛
⎞
–=
x
8
x
6
– x
4
x
2
–1++
hp40g+.book Page 63 Friday, December 9, 2005 1:03 AM