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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

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

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()