diff --git a/src/sage/rings/polynomial/polynomial_element.pyx b/src/sage/rings/polynomial/polynomial_element.pyx
index 9c70fb9fb9f..7860ea029e8 100644
--- a/src/sage/rings/polynomial/polynomial_element.pyx
+++ b/src/sage/rings/polynomial/polynomial_element.pyx
@@ -1485,6 +1485,16 @@ cdef class Polynomial(CommutativeAlgebraElement):
             ...
             AlarmInterrupt
 
+        Check root computation over large finite fields::
+
+            sage: K.<a> = GF(2**50)
+            sage: x = polygen(K)
+            sage: (x**10+x+a).any_root()
+            a^49 + a^47 + a^44 + a^42 + a^41 + a^39 + a^38 + a^37 + a^36 + a^34 + a^33 + a^29 + a^27 + a^26 + a^25 + a^23 + a^18 + a^13 + a^7 + a^5 + a^4 + a^3 + a^2 + a
+            sage: K.<a> = GF(2**150)
+            sage: x = polygen(K)
+            sage: (x**10+x+a).any_root()
+            a^149 + a^148 + a^146 + a^144 + a^143 + a^140 + a^138 + a^136 + a^134 + a^132 + a^131 + a^130 + a^129 + a^127 + a^123 + a^120 + a^118 + a^114 + a^113 + a^112 + a^111 + a^108 + a^104 + a^103 + a^102 + a^99 + a^98 + a^94 + a^91 + a^90 + a^88 + a^79 + a^78 + a^75 + a^73 + a^72 + a^67 + a^65 + a^64 + a^63 + a^62 + a^61 + a^59 + a^57 + a^52 + a^50 + a^48 + a^47 + a^46 + a^45 + a^43 + a^41 + a^39 + a^37 + a^34 + a^31 + a^29 + a^27 + a^25 + a^23 + a^22 + a^20 + a^18 + a^16 + a^14 + a^11 + a^10 + a^8 + a^6 + a^5 + a^4 + a + 1
         """
         if self.base_ring().is_finite() and self.base_ring().is_field():
             if self.degree() < 0: