diff --git a/src/sage/rings/finite_rings/hom_finite_field.pyx b/src/sage/rings/finite_rings/hom_finite_field.pyx
index c816958a81d..bb216a813ea 100644
--- a/src/sage/rings/finite_rings/hom_finite_field.pyx
+++ b/src/sage/rings/finite_rings/hom_finite_field.pyx
@@ -184,6 +184,15 @@ cdef class SectionFiniteFieldHomomorphism_generic(Section):
 cdef class FiniteFieldHomomorphism_generic(RingHomomorphism_im_gens):
     """
     A class implementing embeddings between finite fields.
+
+    TESTS::
+
+        sage: from sage.rings.finite_rings.hom_finite_field import FiniteFieldHomomorphism_generic
+        sage: k.<t> = GF(3^7)
+        sage: K.<T> = GF(3^21)
+        sage: f = FiniteFieldHomomorphism_generic(Hom(k, K))
+        sage: TestSuite(f).run()
+
     """
     def __init__(self, parent, im_gens=None, base_map=None, check=True, section_class=None):
         """
@@ -403,8 +412,68 @@ cdef class FiniteFieldHomomorphism_generic(RingHomomorphism_im_gens):
         return self.section()(b)
 
     def __hash__(self):
+        r"""
+        Return a hash of this morphism
+
+        TESTS::
+
+            sage: k.<t> = GF(5^3)
+            sage: Frob = k.frobenius_endomorphism()
+            sage: embed = Frob.fixed_field()[1]
+            sage: hash(embed)  # random
+            -2441354824160407762 
+        """
         return Morphism.__hash__(self)
 
+    cdef dict _extra_slots(self):
+        r"""
+        Helper function for copying and pickling
+
+        TESTS::
+
+            sage: k.<t> = GF(5^3)
+            sage: Frob = k.frobenius_endomorphism()
+            sage: embed = Frob.fixed_field()[1]
+            sage: embed.__reduce__()  # indirect doctest
+            (<built-in function unpickle_map>,
+             (<class 'sage.rings.finite_rings.hom_prime_finite_field.FiniteFieldHomomorphism_prime'>,
+              Set of field embeddings from Finite Field of size 5 to Finite Field in t of size 5^3,
+              {},
+              {'__im_gens': [1],
+               '_base_map': None,
+               '_codomain': Finite Field in t of size 5^3,
+               '_domain': Finite Field of size 5,
+               '_is_coercion': False,
+               '_lift': None,
+               '_repr_type_str': None,
+               '_section_class': <class 'sage.rings.finite_rings.hom_prime_finite_field.SectionFiniteFieldHomomorphism_prime'>}))
+        """
+        cdef dict slots
+        slots = RingHomomorphism_im_gens._extra_slots(self)
+        slots['_section_class'] = self._section_class
+        return slots
+
+    cdef _update_slots(self, dict slots):
+        r"""
+        Helper function for copying and pickling
+
+        TESTS::
+
+            sage: k.<t> = GF(5^3)
+            sage: Frob = k.frobenius_endomorphism()
+            sage: embed = Frob.fixed_field()[1]
+            sage: f = loads(dumps(embed))
+            sage: f == embed
+            True
+            sage: f.section()
+            Section of Ring morphism:
+              From: Finite Field of size 5
+              To:   Finite Field in t of size 5^3
+              Defn: 1 |--> 1
+        """
+        RingHomomorphism_im_gens._update_slots(self, slots)
+        self._section_class = slots['_section_class']
+
 
 cdef class FrobeniusEndomorphism_finite_field(FrobeniusEndomorphism_generic):
     """