gh-36971: add helper method to concatenate vectors
    
Writing `vector(R, list(v) + list(w))` is a common way to concatenate
two vectors `v` and `w` over some ring `R`.

In this patch we introduce a simple helper method which allows writing
`v.concatenate(w)` instead.
    
URL: #36971
Reported by: Lorenz Panny
Reviewer(s): grhkm21, Ruchit Jagodara

Author: Release Manager

src/sage/modules/free_module_element.pyx

@@ -4148,6 +4148,87 @@ cdef class FreeModuleElement(Vector):   # abstract base class
 
     nintegrate=nintegral
 
+    def concatenate(self, other, *, ring=None):
+        r"""
+        Return the result of concatenating this vector with a sequence
+        of elements given by another iterable.
+
+        If the optional keyword argument ``ring`` is passed, this method
+        will return a vector over the specified ring (or fail). If no
+        base ring is given, the base ring is determined automatically by
+        the :func:`vector` constructor.
+
+        EXAMPLES::
+
+            sage: v = vector([1, 2, 3])
+            sage: w = vector([4, 5])
+            sage: v.concatenate(w)
+            (1, 2, 3, 4, 5)
+            sage: v.parent()
+            Ambient free module of rank 3 over the principal ideal domain Integer Ring
+            sage: w.parent()
+            Ambient free module of rank 2 over the principal ideal domain Integer Ring
+            sage: v.concatenate(w).parent()
+            Ambient free module of rank 5 over the principal ideal domain Integer Ring
+
+        Forcing a base ring is possible using the ``ring`` argument::
+
+            sage: v.concatenate(w, ring=QQ)
+            (1, 2, 3, 4, 5)
+            sage: v.concatenate(w, ring=QQ).parent()
+            Vector space of dimension 5 over Rational Field
+
+        ::
+
+            sage: v.concatenate(w, ring=Zmod(3))
+            (1, 2, 0, 1, 2)
+
+        The method accepts arbitrary iterables of elements which can
+        be coerced to a common base ring::
+
+            sage: v.concatenate(range(4,8))
+            (1, 2, 3, 4, 5, 6, 7)
+            sage: v.concatenate(range(4,8)).parent()
+            Ambient free module of rank 7 over the principal ideal domain Integer Ring
+
+        ::
+
+            sage: w2 = [4, QQbar(-5).sqrt()]
+            sage: v.concatenate(w2)
+            (1, 2, 3, 4, 2.236...*I)
+            sage: v.concatenate(w2).parent()
+            Vector space of dimension 5 over Algebraic Field
+            sage: w2 = vector(w2)
+            sage: v.concatenate(w2)
+            (1, 2, 3, 4, 2.236...*I)
+            sage: v.concatenate(w2).parent()
+            Vector space of dimension 5 over Algebraic Field
+
+        ::
+
+            sage: w2 = polygen(QQ)^4 + 5
+            sage: v.concatenate(w2)
+            (1, 2, 3, 5, 0, 0, 0, 1)
+            sage: v.concatenate(w2).parent()
+            Vector space of dimension 8 over Rational Field
+            sage: v.concatenate(w2, ring=ZZ)
+            (1, 2, 3, 5, 0, 0, 0, 1)
+            sage: v.concatenate(w2, ring=ZZ).parent()
+            Ambient free module of rank 8 over the principal ideal domain Integer Ring
+
+        ::
+
+            sage: v.concatenate(GF(9).gens())
+            (1, 2, 0, z2)
+            sage: v.concatenate(GF(9).gens()).parent()
+            Vector space of dimension 4 over Finite Field in z2 of size 3^2
+        """
+        from itertools import chain
+        coeffs = chain(self, other)
+        if ring is not None:
+            return vector(ring, coeffs)
+        return vector(coeffs)
+
 #############################################
 # Generic dense element
 #############################################