Say that
(
x
,
y
,
z
)
(x, y, z)
is a positive primitive integral Pythagorean triple if
x
,
y
,
z
x, y, z
are positive integers without common factors satisfying
x
2
+
y
2
=
z
2
x^2 + y^2 = z^2
. An old theorem of Berggren gives three integral invertible linear transformations whose semi-group actions on
(
3
,
4
,
5
)
(3, 4, 5)
and
(
4
,
3
,
5
)
(4, 3, 5)
generate all positive primitive Pythagorean triples in a unique manner. We establish an analogue of Berggren’s theorem in the context of a one-variable polynomial ring over a field of characteristic
≠
2
\neq 2
. As its corollaries, we obtain some structure theorems regarding the orthogonal group with respect to the Pythagorean quadratic form over the polynomial ring.