Exercise 9.1.13

Proof.

(a)

By Exercise 12(c), $$\begin{array}{llll}\hfill {\Phi }_{15}(x)& =\frac{{\Phi }_3(x^5)}{{\Phi }_3(x)}& \hfill & \\ \hfill & =\frac{x^{10}+x^5+1}{x^2+x+1}& \hfill & \\ \hfill & =x^8-x^7+x^5-x^4+x^3-x+1.& \hfill & \end{array}$$

(b)

$$\begin{array}{llll}\hfill {\Phi }_{105}(x)& =\frac{{\Phi }_{15}(x^7)}{{\Phi }_{15}(x)}& \hfill & \\ \hfill =& \frac{x^{56}-x^{49}+x^{35}-x^{28}+x^{21}-x^7+1}{x^8-x^7+x^5-x^4+x^3-x+1}& \hfill & \\ \hfill =& x^{48}+x^{47}+x^{46}-x^{43}-x^{42}-2x^{41}-x^{40}-x^{39}+x^{36}+x^{35}+x^{34}+x^{33}+x^{32}& \hfill & \\ \hfill & +x^{31}-x^{28}-x^{26}-x^{24}-x^{22}-x^{20}+x^{17}+x^{16}+x^{15}+x^{14}+x^{13}+x^{12}-x^9-x^8& \hfill & \\ \hfill & -2x^7-x^6-x^5+x^2+x+1& \hfill & \end{array}$$