From ae8cfa3610b600b0d7f27f27599aca5cb86cdd44 Mon Sep 17 00:00:00 2001
From: Travis Scrimshaw <tcscrims@gmail.com>
Date: Tue, 16 Aug 2022 14:54:57 +0900
Subject: [PATCH] Make is_integral_domain() always take a proof keyword
 arugment.

---
 src/sage/categories/integral_domains.py | 15 +++++++++++----
 src/sage/rings/tate_algebra.py          | 13 +++++++++++--
 2 files changed, 22 insertions(+), 6 deletions(-)

diff --git a/src/sage/categories/integral_domains.py b/src/sage/categories/integral_domains.py
index 9daa68ff25e..8cf87ffe94f 100644
--- a/src/sage/categories/integral_domains.py
+++ b/src/sage/categories/integral_domains.py
@@ -96,17 +96,23 @@ def _contains_helper(cls):
         return Category_contains_method_by_parent_class(cls())
 
     class ParentMethods:
-        def is_integral_domain(self):
-            """
-            Return True, since this in an object of the category of integral domains.
+        def is_integral_domain(self, proof=True):
+            r"""
+            Return ``True``, since this in an object of the category
+            of integral domains.
 
             EXAMPLES::
 
                 sage: QQ.is_integral_domain()
                 True
-                sage: Parent(QQ,category=IntegralDomains()).is_integral_domain()
+                sage: Parent(QQ, category=IntegralDomains()).is_integral_domain()
                 True
 
+                sage: L.<z> = LazyLaurentSeriesRing(QQ)
+                sage: L.is_integral_domain()
+                True
+                sage: L.is_integral_domain(proof=True)
+                True
             """
             return True
 
@@ -138,3 +144,4 @@ def _test_fraction_field(self, **options):
 
     class ElementMethods:
         pass
+
diff --git a/src/sage/rings/tate_algebra.py b/src/sage/rings/tate_algebra.py
index fd0147eaf88..9097b2a4e07 100644
--- a/src/sage/rings/tate_algebra.py
+++ b/src/sage/rings/tate_algebra.py
@@ -1275,8 +1275,8 @@ def random_element(self, degree=2, terms=5, integral=False, prec=None):
             gens = [ self.element_class(self, g) for g in self._integer_ring._gens ]
         return self.element_class(self, polring.random_element(degree, terms)(*gens), prec)
 
-    def is_integral_domain(self):
-        """
+    def is_integral_domain(self, proof=True):
+        r"""
         Return ``True`` since any Tate algebra is an integral domain.
 
         EXAMPLES::
@@ -1285,5 +1285,14 @@ def is_integral_domain(self):
             sage: A.is_integral_domain()
             True
 
+        TESTS:
+
+        Check that :trac:`34372` is fixed::
+
+            sage: A.<x,y> = TateAlgebra(Zp(3))
+            sage: R.<a,b> = PolynomialRing(A)
+            sage: R.is_integral_domain(proof=True)
+            True
         """
         return True
+