Factorization tables for trinomials over $\mathrm {GF}(q)$

Abstract:

Tables placed in the UMT file give the complete factorization over ${\text {GF}}(q),q = {p^a}$, of each trinomial $T(x)$ of degree $n, 2 \leqslant n \leqslant d$, as below, together with the generalized Euler $\Phi$-function whenever $T(x)$ is not prime and $\Phi (T(x)) < {10^8}$. In addition, the numerical exponent and q-polynomial is given for each $T(x)$ whenever $2 \leqslant n \leqslant {d_1}$. \[ \begin {array}{*{20}{c}} {q = 2:d = 20,{d_1} = 18,} \hfill & {q = 5:d = 15,{d_1} = 8,} \hfill \\ {q = {2^2}:d = 16,{d_1} = 10,} \hfill & {q = 7:d = 10,{d_1} = 7,} \hfill \\ {q = {2^3}:d = 9,{d_1} = 7,} \hfill & {q = 11:d = 7,} \hfill \\ {q = {2^4}:d = 8,} \hfill & {q = 13:d = 7,} \hfill \\ {q = 3:d = 18,{d_1} = 9,} \hfill & {q = 17:d = 7,} \hfill \\ {q = {3^2}:d = 9,} \hfill & {q = 19:d = 7.} \hfill \\ \end {array} \] On a microfiche card with this note, selected results from the above appear as Table I-Table IV as follows: \[ \begin {array}{*{20}{c}} {q = 2:d = 20,{d_1} = 18,} \hfill & {q = 3:d = 11,{d_1} = 9,} \hfill \\ {q = {2^2}:d = 8,{d_1} = 8,} \hfill & {q = 5:d = 5,{d_1} = 5.} \hfill \\ \end {array} \] As evidenced by these tables, there does not necessarily exist a prime trinomial of given degree n over arbitrary ${\text {GF}}(q)$.