1 | initial version |

Let me first answer the secont part. When you write `PRZp = PolynomialRing(Zp, 'x')`

, you only define `PRZp`

to be a polynomial ring with the symbol `'x'`

as an indeterminate, but you did not let the PYthon variable `x`

to point to the polynomial undeterminate `'x'`

. For this, you can for example do:

```
sage: x = PRZp.gen()
```

or

```
sage: PRZp.inject_variables()
Defining x
```

Now, for the first part, the notation `PRZp.<x> = Zp[]`

is indeed not Pythonic, so it works because of Sage preparser that transforms it to something meaningful to Python:

```
sage: preparse('PRZp.<x> = Zp[]')
"PRZp = Zp['x']; (x,) = PRZp._first_ngens(1)"
```

So, you can either writh that in your python file, or, if you want to keep the same syntax, you should use a Sage file instead ot a Python file, so that Sage preparser get applied first. The way to do this is to rename `yourfile.py`

into `yourfile.sage`

.

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