gh-37085: remove entirely the legacy class EuclideanDomain
    
This is getting rid of the deprecated class `EuclideanDomain`, not used
anywhere.

The element class `EuclideanDomainElement` is kept.

This is one step in cleaning and simplifying the coercion framework.

### 📝 Checklist

- [x] The title is concise, informative, and self-explanatory.
- [x] The description explains in detail what this PR is about.
    
URL: #37085
Reported by: Frédéric Chapoton
Reviewer(s): Peter Bruin

Author: Release Manager

src/sage/rings/abc.pyx

@@ -1,7 +1,7 @@
 """
 Abstract base classes for rings
 """
-from sage.rings.ring import EuclideanDomain
+from sage.rings.ring import IntegralDomain
 
 
 class NumberField_quadratic(Field):
@@ -419,7 +419,7 @@ class Order:
     pass
 
 
-class pAdicRing(EuclideanDomain):
+class pAdicRing(IntegralDomain):
     r"""
     Abstract base class for :class:`~sage.rings.padics.generic_nodes.pAdicRingGeneric`.
 

src/sage/rings/all.py

@@ -16,7 +16,7 @@
 
 # Ring base classes
 from sage.rings.ring import (Ring, Field, CommutativeRing, IntegralDomain,
-    DedekindDomain, PrincipalIdealDomain, EuclideanDomain)
+                             DedekindDomain, PrincipalIdealDomain)
 
 # Ring element base classes
 from sage.structure.element import (CommutativeAlgebraElement,

src/sage/rings/ring.pxd

@@ -23,9 +23,6 @@ cdef class DedekindDomain(IntegralDomain):
 cdef class PrincipalIdealDomain(IntegralDomain):
     pass
 
-cdef class EuclideanDomain(PrincipalIdealDomain):
-    pass
-
 cdef class Field(PrincipalIdealDomain):
     pass
 

src/sage/rings/ring.pyx

@@ -29,7 +29,6 @@ The class inheritance hierarchy is:
 
 Subclasses of :class:`PrincipalIdealDomain` are
 
-- :class:`EuclideanDomain`
 - :class:`Field`
 
   - :class:`~sage.rings.finite_rings.finite_field_base.FiniteField`
@@ -77,7 +76,6 @@ from sage.categories.rings import Rings
 from sage.categories.commutative_rings import CommutativeRings
 from sage.categories.integral_domains import IntegralDomains
 from sage.categories.principal_ideal_domains import PrincipalIdealDomains
-from sage.categories.euclidean_domains import EuclideanDomains
 
 _Rings = Rings()
 _CommutativeRings = CommutativeRings()
@@ -1639,7 +1637,7 @@ cdef class IntegralDomain(CommutativeRing):
         This method is used by all the abstract subclasses of
         :class:`IntegralDomain`, like :class:`NoetherianRing`,
         :class:`PrincipalIdealDomain`, :class:`DedekindDomain`,
-        :class:`EuclideanDomain`, :class:`Field`, ... in order to
+        :class:`Field`, ... in order to
         avoid cascade calls Field.__init__ ->
         PrincipalIdealDomain.__init__ -> IntegralDomain.__init__ ->
         ...
@@ -1654,10 +1652,6 @@ cdef class IntegralDomain(CommutativeRing):
             sage: F.category()
             Category of principal ideal domains
 
-            sage: F = EuclideanDomain(QQ)
-            sage: F.category()
-            Category of euclidean domains
-
             sage: F = Field(QQ)
             sage: F.category()
             Category of fields
@@ -1676,9 +1670,6 @@ cdef class IntegralDomain(CommutativeRing):
             sage: PrincipalIdealDomain._default_category
             Category of principal ideal domains
 
-            sage: EuclideanDomain._default_category
-            Category of euclidean domains
-
             sage: Field._default_category
             Category of fields
 
@@ -2071,27 +2062,6 @@ cdef class PrincipalIdealDomain(IntegralDomain):
         from sage.rings.ideal import Ideal_pid
         return Ideal_pid
 
-cdef class EuclideanDomain(PrincipalIdealDomain):
-    """
-    Generic Euclidean domain class.
-
-    This class is deprecated. Please use the
-    :class:`~sage.categories.euclidean_domains.EuclideanDomains`
-    category instead.
-    """
-    _default_category = EuclideanDomains()
-
-    def parameter(self):
-        """
-        Return an element of degree 1.
-
-        EXAMPLES::
-
-            sage: R.<x>=QQ[]
-            sage: R.parameter()
-            x
-       """
-        raise NotImplementedError
 
 cpdef bint _is_Field(x) except -2:
     """