Let be a nonzero complex number where . Then, let and and define another complex number . Now, let for .
- Prove that . We say is an th root of .
- Prove that has exactly distinct th roots given by
and that they are equally spaced on a circle of radius .
- Find the three cube roots of .
- Find the four fourth roots of .
- Find the four fourth roots of .
- Proof. Using the definitions of and we compute,
- Proof. From part (a) we know is an th root of . Then, if we have
(since for all ). Hence, is an th root of .
Then, if and we have for . This implies
for ; hence, there are distinct values of . Therefore, has exactly distinct th roots. By the Fundamental Theorem of Algebra we know it cannot have more than roots; hence, we have shown that they are all of the form
- For the complex number we have
Therefore, from parts (a) and (b) we have
Hence, the three cube roots of are
- This time we have
Therefore the four fourth roots of are
- For we have
Therefore,
Thus,
To show that the n roots corresponding to powers of epsilon are actually distinct, you can assume that two of them are equal that is (q^k1)z1=(q^k2)z1 where q is epsilon and 0≤k1,k2≤n-1, then since z1≠0 we have q^k1=q^k2 which implies that
exp(i2pi(k1-k2)/n)=1 ==> (k1-k2)/n=m ,an integer, but |k1-k2|0 there are integers t and s such that m=tn+s with 0≤s<n thus a=2tpi+2spi/n+d/n so
z2=(r^1/n)(e^ia)=(r^1/n)(e^i2tpi)(e^i2spi/n)(e^id/n)=
z1q^s which is one of the n roots we found since 0≤s<n.
Something went wrong, hopefully it works this time:
To show that the n roots corresponding to powers of epsilon are actually distinct, you can assume that two of them are equal that is (q^k1)z1=(q^k2)z1 where q is epsilon and 0≤k1,k2≤n-1, then since z1≠0 we have q^k1=q^k2 which implies that
exp(i2pi(k1-k2)/n)=1 ==> (k1-k2)/n=m ,an integer, but abs(k1-k2)0 there are integers t and s such that m=tn+s with 0≤s<n thus a=2tpi+2spi/n+d/n so
z2=(r^1/n)(e^ia)=(r^1/n)(e^i2tpi)(e^i2spi/n)(e^id/n)=
z1q^s which is one of the n roots we found since 0≤s<n.
It got messed up again, basically you take two of them with exponents k1 and k2 and show that k1=k2 using the fact that their difference cannot exceed n-1. To show there aren’t more roots you assume there is one.. you prove it has the same modulus.. you prove the difference of exponents must be 2mpi for some integer m and then use the division algorithm (exercise 10 page 36) to carry out the rest of the proof.