antipalindromic polynomial is antipalindromic. A palindromic polynomial of odd degree is a multiple of (it has –1 as a root) and its quotient by is also palindromic. An antipalindromic polynomial over a field with odd characteristic is a multiple of (it has 1 as a root) and its quotient by is palindromic. An antipalindromic polynomial of even degree is a multiple of (it has −1 and 1 as a roots) and its quotient by is palindromic. If is a palindromic polynomial of even degree 2, then there is a polynomial of degree such that . If is a monic antipalindromic polynomial of even degree 2 over a field of odd characteristic, then it can be written uniquely as , where is a monic polynomial of degree with no constant term. If an antipalindromic polynomial has even degree over a field of odd characteristic, then its "middle" coefficient (of power ) is 0 since . Real coefficients A polynomial with real coefficients all of whose complex roots lie on the unit circle in the complex plane (that is, all the roots have modulus 1) is either palindromic or antipalindromic. Conjugate reciprocal polynomials A polynomial is conjugate reciprocal if and self-inversive if for a scale factor on the unit circle. If is the minimal polynomial of with , and has real coefficients, then is self-reciprocal. This follows because So is a root of the polynomial which has degree . But, the minimal polynomial is unique, hence for some constant , i.e. . Sum from to and note that 1 is not a root of . We conclude that . A consequence is that the cyclotomic polynomials are self-reciprocal for . This is used in the special number field sieve to allow numbers of the form and to be factored taking advantage of the algebraic factors by using polynomials of degree 5, 6, 4 and 6 respectively – note that (Euler's totient function) of the exponents are 10, 12, 8 and 12. Per Cohn's theorem, a self-inversive polynomial has as many roots in the unit disk as the reciprocal polynomial of its derivative. Application in coding theory The reciprocal polynomial finds a use in the theory of cyclic error correcting codes. Suppose can be factored into the product of two polynomials, say . When generates a cyclic code , then the reciprocal polynomial generates , the orthogonal complement of . Also, is self-orthogonal