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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()





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