Editing Cyclotomic polynomial (section)

===Gauss's formula===

Let ''n'' be odd, [[Square-free integer|square-free]], and greater than 3. Then:<ref>Gauss, DA, Articles 356-357</ref><ref name=riesel>{{citation
 | last1 = Riesel  | first1 = Hans
 | title = Prime Numbers and Computer Methods for Factorization
 | edition = 2nd
 | publisher = Birkhäuser
 | location = Boston
 | year = 1994
 | isbn = 0-8176-3743-5
 | pages = 309-316, 436, 443
}}</ref>

:<math>4\Phi_n(z) = A_n^2(z) - (-1)^{\frac{n-1}{2}}nz^2B_n^2(z)</math>

for certain polynomials ''A<sub>n</sub>''(''z'') and ''B<sub>n</sub>''(''z'') with integer coefficients, ''A<sub>n</sub>''(''z'') of degree ''&phi;''(''n'')/2, and ''B<sub>n</sub>''(''z'') of degree ''&phi;''(''n'')/2 − 2. Furthermore, ''A<sub>n</sub>''(''z'') is palindromic when its degree is even; if its degree is odd it is antipalindromic. Similarly, ''B<sub>n</sub>''(''z'') is palindromic unless ''n'' is composite and ''n'' ≡ 3 (mod 4), in which case it is antipalindromic.

The first few cases are

:<math>\begin{align}
4\Phi_5(z) &=4(z^4+z^3+z^2+z+1)\\ 
&= (2z^2+z+2)^2 - 5z^2  \\[6pt]
4\Phi_7(z) &=4(z^6+z^5+z^4+z^3+z^2+z+1)\\ 
&= (2z^3+z^2-z-2)^2+7z^2(z+1)^2 \\ [6pt]
4\Phi_{11}(z)
&=4(z^{10}+z^9+z^8+z^7+z^6+z^5+z^4+z^3+z^2+z+1)\\ 
&= (2z^5+z^4-2z^3+2z^2-z-2)^2+11z^2(z^3+1)^2
\end{align}</math>