From 2fe12dce586d0023a326262fe82b8d43f25b1f9e Mon Sep 17 00:00:00 2001
From: Hetarth Jodha <hetarthj7@gmail.com>
Date: Fri, 28 Nov 2025 13:49:02 +0530
Subject: [PATCH 1/2] Fix #41222: Implement radical() for 0 in Integer ring

---
 src/sage/rings/integer.pyx | 19 +++++++++++++++++++
 1 file changed, 19 insertions(+)

diff --git a/src/sage/rings/integer.pyx b/src/sage/rings/integer.pyx
index bdd2a7ce05b..3be2d036752 100644
--- a/src/sage/rings/integer.pyx
+++ b/src/sage/rings/integer.pyx
@@ -5874,6 +5874,25 @@ cdef class Integer(sage.structure.element.EuclideanDomainElement):
         flag = self < 0 and proof
         return objtogen(self).qfbclassno(flag).sage()
 
+    def radical(self):
+        """
+        Return the radical of this integer, i.e., the product of its
+        prime divisors.
+
+        EXAMPLES::
+
+            sage: 0.radical()
+            0
+            sage: 10.radical()
+            10
+            sage: (-12).radical()
+            6
+        """
+        if self.is_zero():
+            return self
+
+        return self.factor().radical_value()
+
     def squarefree_part(self, long bound=-1):
         r"""
         Return the square free part of `x` (=``self``), i.e., the unique integer

From ad8b1474a5205de02147c4cf51eb27e5bb13dedd Mon Sep 17 00:00:00 2001
From: Hetarth Jodha <hetarthj7@gmail.com>
Date: Fri, 28 Nov 2025 21:39:35 +0530
Subject: [PATCH 2/2] Fix signature of radical() to accept args, and update
 doctests

---
 src/sage/arith/misc.py                              | 4 +---
 src/sage/categories/unique_factorization_domains.py | 4 +---
 src/sage/rings/integer.pyx                          | 6 +++---
 3 files changed, 5 insertions(+), 9 deletions(-)

diff --git a/src/sage/arith/misc.py b/src/sage/arith/misc.py
index 9fe86a04c6b..6969cf26cc5 100644
--- a/src/sage/arith/misc.py
+++ b/src/sage/arith/misc.py
@@ -2775,9 +2775,7 @@ def radical(n, *args, **kwds):
         sage: radical(2 * 3^2 * 5^5)
         30
         sage: radical(0)
-        Traceback (most recent call last):
-        ...
-        ArithmeticError: radical of 0 is not defined
+        0
         sage: K.<i> = QuadraticField(-1)                                                # needs sage.rings.number_field
         sage: radical(K(2))                                                             # needs sage.rings.number_field
         i - 1
diff --git a/src/sage/categories/unique_factorization_domains.py b/src/sage/categories/unique_factorization_domains.py
index f01763d4a7e..ee534d5e3fe 100644
--- a/src/sage/categories/unique_factorization_domains.py
+++ b/src/sage/categories/unique_factorization_domains.py
@@ -262,9 +262,7 @@ def radical(self, *args, **kwds):
                 sage: Integer(-100).radical()
                 10
                 sage: Integer(0).radical()
-                Traceback (most recent call last):
-                ...
-                ArithmeticError: radical of 0 is not defined
+                0
 
             The next example shows how to compute the radical of a number,
             assuming no prime > 100000 has exponent > 1 in the factorization::
diff --git a/src/sage/rings/integer.pyx b/src/sage/rings/integer.pyx
index 3be2d036752..de732516d1e 100644
--- a/src/sage/rings/integer.pyx
+++ b/src/sage/rings/integer.pyx
@@ -5874,7 +5874,7 @@ cdef class Integer(sage.structure.element.EuclideanDomainElement):
         flag = self < 0 and proof
         return objtogen(self).qfbclassno(flag).sage()
 
-    def radical(self):
+    def radical(self, *args, **kwds):
         """
         Return the radical of this integer, i.e., the product of its
         prime divisors.
@@ -5890,8 +5890,8 @@ cdef class Integer(sage.structure.element.EuclideanDomainElement):
         """
         if self.is_zero():
             return self
-
-        return self.factor().radical_value()
+        
+        return self.factor(*args, **kwds).radical_value()
 
     def squarefree_part(self, long bound=-1):
         r"""