diff --git a/qualtran/_infra/data_types.py b/qualtran/_infra/data_types.py
index 8c541bfcc..bc3e80e28 100644
--- a/qualtran/_infra/data_types.py
+++ b/qualtran/_infra/data_types.py
@@ -51,7 +51,7 @@
 import abc
 from enum import Enum
 from functools import cached_property
-from typing import Any, Iterable, List, Optional, Sequence, TYPE_CHECKING, Union
+from typing import Any, Iterable, List, Literal, Optional, Sequence, TYPE_CHECKING, Union
 
 import attrs
 import numpy as np
@@ -906,8 +906,19 @@ class QGF(QDType):
 
     characteristic: SymbolicInt
     degree: SymbolicInt
-    irreducible_poly: Optional['galois.Poly'] = None
-    element_repr: str = 'int'
+    irreducible_poly: Optional['galois.Poly'] = attrs.field()
+    element_repr: Literal["int", "poly", "power"] = attrs.field(default='int')
+
+    @irreducible_poly.default
+    def _irreducible_poly_default(self):
+        if is_symbolic(self.characteristic, self.degree):
+            return None
+
+        from galois import GF
+
+        return GF(  # type: ignore[call-overload]
+            int(self.characteristic), int(self.degree), compile='python-calculate'
+        ).irreducible_poly
 
     @cached_property
     def order(self) -> SymbolicInt:
@@ -936,10 +947,12 @@ def _quint_equivalent(self) -> QUInt:
     def gf_type(self):
         from galois import GF
 
+        poly = self.irreducible_poly if self.degree > 1 else None
+
         return GF(  # type: ignore[call-overload]
             int(self.characteristic),
             int(self.degree),
-            irreducible_poly=self.irreducible_poly,
+            irreducible_poly=poly,
             repr=self.element_repr,
             compile='python-calculate',
         )
diff --git a/qualtran/_infra/data_types_test.py b/qualtran/_infra/data_types_test.py
index ff08b7df6..0c889cdc8 100644
--- a/qualtran/_infra/data_types_test.py
+++ b/qualtran/_infra/data_types_test.py
@@ -540,3 +540,11 @@ def test_montgomery_bit_conversion(bitsize):
     dtype = QMontgomeryUInt(bitsize)
     for v in range(1 << bitsize):
         assert v == dtype.from_bits(dtype.to_bits(v))
+
+
+def test_qgf_with_default_poly_is_compatible():
+    qgf_one = QGF(2, 4)
+
+    qgf_two = QGF(2, 4, irreducible_poly=qgf_one.gf_type.irreducible_poly)
+
+    assert qgf_one == qgf_two