sagemathgh-39583: Handle various extension degrees in pAdicGenericElement initialisation
    
I have looked into bug sagemath#28555. I have been able to reproduce it with
version 10.6.beta7.

It turns out the infinite loop can be reproduced with:
```
sage: A.<a> = Qq(5^2)
sage: A.base_ring()(a)
```
But the following also triggers the bug: `A.base_ring()(A(1))`. This is
bug sagemath#33527.

The culprit is in the file
`src/sage/rings/padics/padic_template_element.pxi`. Calling
`A.base_ring()(a)` will call `pAdicTemplateElement.__init__` which will
call `A.base_ring()(a)` again on line 140.

To fix it, I added a check for when the degree of the polynomial (over
the base ring $\mathbb Z_p$ or $\mathbb Q_p$) defining the element is
greater than one. If we were trying to convert to the base ring, I raise
a `TypeError` exception. Else, I raise a `NotImplementedError` exception
(this is the case when one tries to convert between two extensions, it
should be implemented later, but here I only fix the bug).

Finally, I change line 140 from `x = self.base_ring()(x)` to `x =
self.base_ring()(x.polynomial().constant_coefficient())` to avoid the
infinite loop.

NOTE: this is my "hello world" PR, I may have broken something with
this, please be kind. Please do not hesitate to tell me how to do
better.

Fixes sagemath#28555.
Fixes sagemath#33527.

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None
    
URL: sagemath#39583
Reported by: Rubén Muñoz--Bertrand
Reviewer(s): Julian Rüth, roed314, Rubén Muñoz--Bertrand

Author: Release Manager

src/sage/rings/padics/padic_template_element.pxi

@@ -116,6 +116,23 @@ cdef class pAdicTemplateElement(pAdicGenericElement):
             Traceback (most recent call last):
             ...
             TypeError: no conversion between padics when prime numbers differ
+
+        Check that bug :issue:`28555` is fixed::
+
+            sage: A.<a> = Qq(5^2)
+            sage: A.base_ring()(A(1))
+            1 + O(5^20)
+            sage: A.base_ring()(a)
+            Traceback (most recent call last):
+            ...
+            TypeError: element in a proper extension
+
+        Check that bug :issue:`33527` is fixed::
+
+            sage: K = Qq(25, names='a')
+            sage: K0 = K.base_ring()
+            sage: K0(K(1))
+            1 + O(5^20)
         """
         self.prime_pow = <PowComputer_?>parent.prime_pow
         pAdicGenericElement.__init__(self, parent)
@@ -137,7 +154,11 @@ cdef class pAdicTemplateElement(pAdicGenericElement):
                 if x.parent().modulus().change_ring(self.base_ring()) == self.parent().modulus():
                     x = x.polynomial().change_ring(self.base_ring()).list()
                 else:
-                    x = self.base_ring()(x)
+                    if x.polynomial().degree() >= 1:
+                        if self.parent() is x.parent().base_ring():
+                            raise TypeError("element in a proper extension")
+                        raise NotImplementedError("conversion between different p-adic extensions not implemented")
+                    x = self.base_ring()(x.polynomial().constant_coefficient())
                     if x.is_zero():
                         absprec = min(absprec, x.precision_absolute()*self.prime_pow.e)
                         x = []