Factorization tables for trinomials over 𝐺𝐹(𝑞)
Authors: Beard, Jr. Jacob T. B. , West Karen I. 
Published: June 2010

Tables placed in the UMT file give the complete factorization over

GF

(
q
)
,
q
=

p
a

{\text {GF}}(q),q = {p^a}

, of each trinomial

T
(
x
)

T(x)

of degree

n
,
2
⩽
n
⩽
d

n, 2 \leqslant n \leqslant d

, as below, together with the generalized Euler

Φ
\Phi

-function whenever

T
(
x
)

T(x)

is not prime and

Φ
(
T
(
x
)
)
>

10
8

\Phi (T(x)) > {10^8}

. In addition, the numerical exponent and q-polynomial is given for each

T
(
x
)

T(x)

whenever

2
⩽
n
⩽

d
1

2 \leqslant n \leqslant {d_1}

.
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row 1st Column q equals 2 colon d equals 20 comma d 1 equals 18 comma 2nd Column a m p semicolon q equals 5 colon d equals 15 comma d 1 equals 8 comma 2nd Row 1st Column q equals 2 squared colon d equals 16 comma d 1 equals 10 comma 2nd Column a m p semicolon q equals 7 colon d equals 10 comma d 1 equals 7 comma 3rd Row 1st Column q equals 2 cubed colon d equals 9 comma d 1 equals 7 comma 2nd Column a m p semicolon q equals 11 colon d equals 7 comma 4th Row 1st Column q equals 2 Superscript 4 Baseline colon d equals 8 comma 2nd Column a m p semicolon q equals 13 colon d equals 7 comma 5th Row 1st Column q equals 3 colon d equals 18 comma d 1 equals 9 comma 2nd Column a m p semicolon q equals 17 colon d equals 7 comma 6th Row 1st Column q equals 3 squared colon d equals 9 comma 2nd Column a m p semicolon q equals 19 colon d equals 7 period EndLayout">

q
=
2
:
d
=
20
,

d
1

=
18
,

a
m
p
;

q
=
5
:
d
=
15
,

d
1

=
8
,

q
=

2
2

:
d
=
16
,

d
1

=
10
,

a
m
p
;

q
=
7
:
d
=
10
,

d
1

=
7
,

q
=

2
3

:
d
=
9
,

d
1

=
7
,

a
m
p
;

q
=
11
:
d
=
7
,

q
=

2
4

:
d
=
8
,

a
m
p
;

q
=
13
:
d
=
7
,

q
=
3
:
d
=
18
,

d
1

=
9
,

a
m
p
;

q
=
17
:
d
=
7
,

q
=

3
2

:
d
=
9
,

a
m
p
;

q
=
19
:
d
=
7.

\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:
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row 1st Column q equals 2 colon d equals 20 comma d 1 equals 18 comma 2nd Column a m p semicolon q equals 3 colon d equals 11 comma d 1 equals 9 comma 2nd Row 1st Column q equals 2 squared colon d equals 8 comma d 1 equals 8 comma 2nd Column a m p semicolon q equals 5 colon d equals 5 comma d 1 equals 5 period EndLayout">

q
=
2
:
d
=
20
,

d
1

=
18
,

a
m
p
;

q
=
3
:
d
=
11
,

d
1

=
9
,

q
=

2
2

:
d
=
8
,

d
1

=
8
,

a
m
p
;

q
=
5
:
d
=
5
,

d
1

=
5.

\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

GF

(
q
)

{\text {GF}}(q)

.