I'm beginning to learn to use SINGULAR, the computer algebra system (CAS) for commutative algebra.
NOTATION: If $K$ is a field of characteristic $0$, then $\mathbb{Q}\subseteq K$; otherwise $\mathbb{Z}_p\!\subseteq\!K$ for some prime $p\!=\!\mathrm{char}\,K$. If $S\subseteq\mathbb{Q}[x_1,\ldots,x_n]\!=\!\mathbb{Q}[\mathbf{x}]\subseteq K[x_1,\ldots,x_n]\!=\!K[\mathbf{x}]$, let $\langle S\rangle_{\mathbb{Q}[\mathbf{x}]}$ denote the ideal in $\mathbb{Q}[\mathbf{x}]$ generated by polinomials from $S$; and let $\langle S\rangle_{K[\mathbf{x}]}$ denote the ideal in $K[\mathbf{x}]$ generated by polinomials from $S$.
From what I understand, CASs are able to carry out computations in $K[\mathbf{x}]$ only when $K$ is either $\mathbb{Q}$ or finite (and some other special cases).
QUESTION: when doing the standard operations with a CAS on ideals, such as intersection, multiplication, addition, intersection with subrings (elimination of variables), etc. within $\mathbb{Q}[\mathbf{x}]$, how much can we deduce about the corresponding ideals in $K[\mathbf{x}]$?
More concretely:
- does $\langle S_1\rangle_{\mathbb{Q}[\mathbf{x}]}=\langle S_2\rangle_{\mathbb{Q}[\mathbf{x}]}$ imply $\langle S_1\rangle_{K[\mathbf{x}]}=\langle S_2\rangle_{K[\mathbf{x}]}$?
- does $\langle S_1\rangle_{\mathbb{Q}[\mathbf{x}]}\cap\langle S_2\rangle_{\mathbb{Q}[\mathbf{x}]}=\langle S\rangle_{\mathbb{Q}[\mathbf{x}]}$ imply $\langle S_1\rangle_{K[\mathbf{x}]}\cap\langle S_2\rangle_{K[\mathbf{x}]}=\langle S\rangle_{K[\mathbf{x}]}$?
- does $\langle S_1\rangle_{\mathbb{Q}[\mathbf{x}]}\;\cdot\;\langle S_2\rangle_{\mathbb{Q}[\mathbf{x}]}=\langle S\rangle_{\mathbb{Q}[\mathbf{x}]}$ imply $\langle S_1\rangle_{K[\mathbf{x}]}\;\cdot\;\langle S_2\rangle_{K[\mathbf{x}]}=\langle S\rangle_{K[\mathbf{x}]}$?
- does $\langle S_1\rangle_{\mathbb{Q}[\mathbf{x}]}+\langle S_2\rangle_{\mathbb{Q}[\mathbf{x}]}=\langle S\rangle_{\mathbb{Q}[\mathbf{x}]}$ imply $\langle S_1\rangle_{K[\mathbf{x}]}+\langle S_2\rangle_{K[\mathbf{x}]}=\langle S\rangle_{K[\mathbf{x}]}$?
- for $f\!\in\!\mathbb{Q}[\mathbf{x}]$, does $f\!\in\!\langle S\rangle_{\mathbb{Q}[\mathbf{x}]}\Leftrightarrow f\!\in\!\langle S\rangle_{K[\mathbf{x}]}$ hold?
- etc. (Elimination of variables, Zariski Closure of the Image, Solving Polynomial Equations, Radical Membership, Quotient of Ideals, ...)
Basically what I'm asking is how can we use Singular (and other CASs) for computation in polynomial rings over an arbitrary field (but all polynomials have coefficients in $\mathbb{Q}$).
My main literature is A Singular Introduction to Commutative Algebra (the authors of the book are the developers of Singular). I would think these questions would be adressed somewhere in that book, but I was not able to find such propositions (I just began reading it). Anyone know where to look?
P.S. if S\!\subseteq\!R'\!\leq\!R, where R',R are commutative rings with 1, does \langle S\rangle_{R'}=\langle S\rangle_R\cap R' hold?