6. (8 points) We saw that William Rowan Hamilton\'s quaternions generalized comp

Question

6. (8 points) We saw that William Rowan Hamilton's quaternions generalized complex numbers, with the caveat that multiplication isn't always commutative (see Burton p. 638). Real numbers will commute with anything, but the values i, j, k don't commute with each other. So 2-i-i- 2, but j.ifi.j. One peculiar property is that the number of roots of a polynomial can exceed its degree. For example, the polynomial x2 + 1 h .g., ti,?±k. Show that this polynonnial has infinitely many roots in the quaternions. (Hint: Show that as long as c2 d21, the quaternion bi cj dk will be a root of r2 +1.) as at least Six rootS, e 01 -C

Explanation / Answer

x2 +1 = 0

Take x = bi + cj + dk

x2 +1 = (bi + cj + dk)(bi + cj + dk) +1 = i2b2 + bc ij + bdik + cbji + cdjk + j2c2 + dbki + dckj + k2d2 +1 = - (b2 + c2 +c2 ) + 1 (using ji = -ij and jk = -kj and ik = -ki)

x2 +1 = 0 =>  - (b2 + c2 +c2 ) + 1 = 0 => (b2 + c2 +c2 ) = 1

Clearly we can choose infinite b , c and d such that above condition holds , hence x2 +1 = 0 has infinite roots

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