sagemathgh-40273: Avoid PariError on finite field construction
    
previously the following raise error

```
            sage: F.<I> = GF((2^128+51)^2); F
```

now it doesn't.


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- [x] The description explains in detail what this PR is about.
- [ ] I have linked a relevant issue or discussion.
- [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: sagemath#40273
Reported by: user202729
Reviewer(s): Frédéric Chapoton

Author: Release Manager

src/sage/rings/finite_rings/element_pari_ffelt.pyx

@@ -538,8 +538,17 @@ cdef class FiniteFieldElement_pari_ffelt(FinitePolyExtElement):
             sage: k.<c> = GF(3^17, impl='pari_ffelt')
             sage: c^20  # indirect doctest
             c^4 + 2*c^3
+
+        TESTS::
+
+            sage: F.<I> = GF((2^128+51)^2)
+            sage: I
+            I
         """
-        return str(new_gen_noclear(self.val))
+        s = str(new_gen_noclear(self.val))
+        if self._parent._need_replace_varname:
+            s = s.replace("x", self._parent._names[0])  # .variable_name() is slower
+        return s
 
     def __hash__(self):
         """
@@ -1304,7 +1313,17 @@ cdef class FiniteFieldElement_pari_ffelt(FinitePolyExtElement):
             sage: b = a**2 + 2*a + 1
             sage: b.__pari__()
             a^2 + 2*a + 1
+
+        TESTS::
+
+            sage: F.<I> = GF((2^128+51)^2)
+            sage: pari(I)
+            Traceback (most recent call last):
+            ...
+            ValueError: variable name illegal in PARI
         """
+        if self._parent._need_replace_varname:
+            raise ValueError("variable name illegal in PARI")
         return new_gen_noclear(self.val)
 
     def _pari_init_(self):

src/sage/rings/finite_rings/finite_field_pari_ffelt.py

@@ -17,6 +17,7 @@
 #*****************************************************************************
 
 
+from cypari2.handle_error import PariError
 from .element_pari_ffelt import FiniteFieldElement_pari_ffelt
 from .finite_field_base import FiniteField
 from .finite_field_constructor import GF
@@ -110,6 +111,13 @@ def __init__(self, p, modulus, name=None):
             sage: R.<x> = PolynomialRing(GF(3))
             sage: k = FiniteField_pari_ffelt(3, x^2 + 2*x + 2, 'a'); k
             Finite Field in a of size 3^2
+
+        TESTS::
+
+            sage: F.<I> = GF((2^128+51)^2); F
+            Finite Field in I of size 340282366920938463463374607431768211507^2
+            sage: type(F)
+            <class 'sage.rings.finite_rings.finite_field_pari_ffelt.FiniteField_pari_ffelt_with_category'>
         """
         n = modulus.degree()
         if n < 2:
@@ -120,7 +128,13 @@ def __init__(self, p, modulus, name=None):
         self._modulus = modulus
         self._degree = n
 
-        self._gen_pari = modulus._pari_with_name(self._names[0]).ffgen()
+        self._need_replace_varname = False
+        try:
+            modulus_pari = modulus._pari_with_name(self.variable_name())
+        except PariError:
+            self._need_replace_varname = True
+            modulus_pari = modulus._pari_with_name()
+        self._gen_pari = modulus_pari.ffgen()
         self._zero_element = self.element_class(self, 0)
         self._one_element = self.element_class(self, 1)
         self._gen = self.element_class(self, self._gen_pari)

src/sage/rings/polynomial/polynomial_element.pyx

@@ -7150,6 +7150,10 @@ cdef class Polynomial(CommutativePolynomial):
         Return polynomial as a PARI object with topmost variable
         ``name``.  By default, use 'x' for the variable name.
 
+        It is not recommended to pass a user-specified value
+        as the variable name, because some names such as ``I``
+        are reserved. See :issue:`20631`, :issue:`40273`.
+
         For internal use only.
 
         EXAMPLES::
@@ -7159,6 +7163,13 @@ cdef class Polynomial(CommutativePolynomial):
             2*x^2 + x
             sage: (2*a^2 + a)._pari_with_name('y')                                      # needs sage.libs.pari
             2*y^2 + y
+
+        TESTS::
+
+            sage: (2*a^2 + a)._pari_with_name('I')                                      # needs sage.libs.pari
+            Traceback (most recent call last):
+            ...
+            cypari2.handle_error.PariError: I already exists with incompatible valence
         """
         vals = [x.__pari__() for x in self.list()]
         return pari(vals).Polrev(name)