Note: by scalar, I am referring to the notion which is very commonly mentioned in tensor algebra (scalar = 0-order tensor); I do not have a definition, so please share it if you have one. Roughly speaking, I sometimes read that a 0-tensor, i.e. a scalar, defines 1 amplitude and 0 directions.
In the general case, a scalar field, e.g. a temperature field $T:\mathbb{R}^3\to \mathbb{R}$ depends on a change of basis: for some invertible $P$, $T(x)\neq T(Px)$.
On the contrary, the determinant, e.g. $\det: M_n(\mathbb{R})\to \mathbb{R}$ is invariant by change of basis: $\det(M)=\det(PMP^{-1})$. Does it mean the determinant is not a scalar?
No, the determinant is a pseudoscalar: not a differential form of degree zero, but of top degree.