gh-39050: Fix exactify caching when interrupted
    
Fix a certain `AttributeError` — if you run the newly added test under
previous version of SageMath then it will give an `AttributeError` in
the last line instead.

(ideally you'd want some polynomial where `exactify()` takes exactly 1
second, use `alarm(0.5)` in first case and no `alarm()` in the second
case to get the correct result — but finding such a polynomial is not
easy, and timing result varies over machines/versions anyway)

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- [x] The description explains in detail what this PR is about.
- [x] I have linked a relevant issue or discussion. (not aware of one)
- [x] I have created tests covering the changes.
- [x] I have updated the documentation and checked the documentation
preview.

### ⌛ Dependencies

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URL: #39050
Reported by: user202729
Reviewer(s): Travis Scrimshaw

Author: Release Manager

src/sage/rings/qqbar.py

@@ -7098,12 +7098,26 @@ def exactify(self):
             sage: cp.exactify()
             sage: cp._exact
             True
+
+        TESTS:
+
+        Check that interrupting ``exactify()`` does not lead to incoherent state::
+
+            sage: x = polygen(AA)
+            sage: p = AA(2)^(1/100) * x + AA(3)^(1/100)
+            sage: cp = AA.common_polynomial(p)
+            sage: alarm(0.5); cp.generator()
+            Traceback (most recent call last):
+            ...
+            AlarmInterrupt
+            sage: alarm(0.5); cp.generator()
+            Traceback (most recent call last):
+            ...
+            AlarmInterrupt
         """
         if self._exact:
             return
 
-        self._exact = True
-
         if self._poly.base_ring() is QQ:
             self._factors = [fac_exp[0] for fac_exp in self._poly.factor()]
             self._gen = qq_generator
@@ -7128,6 +7142,8 @@ def exactify(self):
 
                 self._factors = [fac_exp[0] for fac_exp in fp.factor()]
 
+        self._exact = True
+
     def factors(self):
         r"""
         EXAMPLES::