1 | initial version |

Unless you declare `x`

as the indeterminate in `R`

,
it is still a "symbolic variable" in the "symbolic ring".

```
sage: x.parent()
Symbolic Ring
sage: p = (x - 1) * (x - 2)
sage: p.parent()
Symbolic Ring
```

Declare `x`

as the indeterminate in `R`

.

```
sage: R = PolynomialRing(ZZ, 'x')
sage: x = R.gen()
```

Then:

```
sage: x.parent()
Univariate Polynomial Ring in x over Integer Ring
sage: p = (x - 1) * (x - 2)
sage: p.parent()
Univariate Polynomial Ring in x over Integer Ring
```

To get the degree, use the `degree`

method:

```
sage: p.degree()
2
```

2 | No.2 Revision |

~~Unless you declare ~~Polynomials (and symbolic expressions) have a `degree`

method.

Beware that even after defining `R`

as in the question,
`x`

~~as the indeterminate in ~~is still a "symbolic variable" in the "symbolic ring".`R`

,
it

In a fresh Sage session:

```
sage: R = PolynomialRing(ZZ, 'x') # defines R but not x
sage: x.parent()
Symbolic Ring
```~~sage: p ~~
sage: q = (x - 1) * (x - 2)
sage: ~~p.parent()
~~q
(x - 1)*(x - 2)
sage: q.parent()
Symbolic Ring

Symbolic expressions have a `degree`

method that gives
the degree with respect to any chosen variable.

```
sage: q.degree(x)
2
```

Declare `x`

as the indeterminate in `R`

.

~~sage: R = PolynomialRing(ZZ, 'x')
~~sage: x = R.gen()

Then:

```
sage: x.parent()
Univariate Polynomial Ring in x over Integer Ring
sage: p = (x - 1) * (x - 2)
sage: p
x^2 - 3*x + 2
sage: p.parent()
Univariate Polynomial Ring in x over Integer Ring
```

Or:

```
sage: p = R(q)
sage: p
x^2 - 3*x + 2
sage: p.parent()
Univariate Polynomial Ring in x over Integer Ring
```

To get the degree, use the `degree`

~~method:~~method (no need
to specify that it is with respect to `x`

now, since
`p`

is a univariate polynomial):

```
sage: p.degree()
2
```

Sage slightly extends Python's syntax to enable defining
`R`

and `x`

at once. For example:

```
sage: R.<x> = PolynomialRing(ZZ)
```

is equivalent to:

```
sage: R = PolynomialRing(ZZ, 'x')
sage: x = R.gen()
```

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