# Notations concerning polynomials over and extensions of rings

I find it hard to keep overview over the notations for sets of polynomials, evaluations of polynomials, and extensions of rings and so on.

Let $$R$$ be a ring, and $$E/R$$ an extension of $$R$$.

Occasionally one finds definitions like this

$$\boxed{R(\alpha) = \{ a_0 + a_1\alpha\ |\ a_i \in R\}}$$

for a specific number $$\alpha \in E/R$$ , i.e. $$R(\alpha) \subseteq E/R$$ is another set of numbers, for example in Lemmermeyer's Quadratische Zahlkörper: On the other side, the ring of polynomials over $$R$$ is defined by

$$\boxed{R[X] = \{a_0 + a_1X + \dots + a_nX^n\ |\ n \in \mathbb{N}, a_i\in R\}}$$

with a unspecific variable $$X$$, i.e. $$R[X]$$ is a set of "open" algebraic expressions.

With

$$\boxed{R^k[X] = \{a_0 + a_1X + \dots + a_kX^k\ |\ a_i\in R\}}$$

one can write

$$\boxed{R[X] = \bigcup_{k=0}^{k< \infty}R^k[X]}$$

Next to $$R[X]$$ we find the definition of

$$\boxed{R(X) = \Big\{ \frac{P}{Q}\ |\ P, Q \in R[X], Q \neq 0\Big\}}$$

If $$R[X]$$ is a set of algebraic expressions (with a variable $$X$$), $$R[\alpha] = R[X \rightarrow \alpha]$$ is a set of numbers by setting the variable $$X$$ to the numerical value $$\alpha$$:

$$\boxed{R[\alpha] = \{a_0 + a_1\alpha^1 + \dots + a_n\alpha^n\ |\ n \in \mathbb{N}, a_i\in R\}}$$

or

$$\boxed{R[\alpha] = \{P(X\rightarrow \alpha)\ |\ P \in R[X]\}}$$

Accordingly:

$$\boxed{R[\alpha] = \bigcup_{k=0}^{k < \infty}R^k[\alpha]}$$

Occasionally:

$$\boxed{R[\alpha] = R^1[\alpha] = R(\alpha)}$$

e.g. for $$R = \mathbb{Q}$$, $$\alpha=\sqrt{2}$$ (more generally: $$\alpha = \sqrt{d}$$, $$d$$ square-free). [Thanks to user Servaes.]

Things would be easier if a consistent notation was used, especially $$R^1[\alpha]$$ instead of $$R(\alpha)$$ for the set $$\{ a_0 + a_1\alpha\ |\ a_i \in R\}$$.

Finally, there is one notation I'm desperately missing: for the minimal extension of a ring $$R$$ that contains all roots of a polynomial $$P(X) \in R[X]$$:

$$\boxed{R\langle P\rangle = R[\rho_1]\dots[\rho_k]}$$

for the roots $$\rho_i$$ of $$P(X)$$, i.e. $$P(\rho_i) = 0$$. It's essential (or isn't it?) that

$$\boxed{R[\rho_1]\dots[\rho_k] = R[\pi(\rho_1)]\dots[\pi(\rho_k)]}$$

for any permutation $$\pi$$ of the roots.

Note that $$\mathbb{Z}[\frac{1}{2}]$$ is an infinite-dimensional vector space over $$\mathbb{Z}$$ (with base $$\{1,\frac{1}{2},\frac{1}{4},\frac{1}{8},\dots\}$$, while $$\mathbb{Z}^1[\sqrt{2}] = \mathbb{Z}[\sqrt{2}]$$ is a two-dimensional one (with base $$\{1,\sqrt{2}\}$$) - like $$\mathbb{Z}[\sqrt{-1}]$$ (with base $$\{1,i\}$$)!

My question is:

Are there attempts to unify (= optimize) notation in the context of polynomials, evaluation of polynomials, and extensions of rings and fields that minimize confusion (that occasionally arises with established notation)?

• Note that in the case of a squarefree integer $m$ we have $$\Bbb{Q}(\sqrt{m})=\Bbb{Q}[\sqrt{m}]=\Bbb{Q}^1[\sqrt{m}],$$ hence the notation is entirely consistent. As for the minimal extension containing all roots of a polynomial; I've often seen the notation $\Omega_P^R$ used in the context of field theory, but I guess it works just as well for rings. – Servaes Nov 3 '18 at 16:17
• "You've seen": just occasionally? Isn't there an established notation? – Hans-Peter Stricker Nov 3 '18 at 16:31
• ...do you have a question? – Eric Wofsey Nov 3 '18 at 17:01
• @EricWofsey: Very good question! I forgot about it. – Hans-Peter Stricker Nov 3 '18 at 17:03
• Each math author is welcome to define a notation (providing a definition) that "optimizes" the discussion of a topic. At the beginning of your post you have "$R(\alpha) = \{a_0 + a_1 \alpha \mid a_0,a_1 \in R \}$" which already involves some abbreviation that might cause confusion. This construction of a ring $R(\alpha)$ depends on some overring $S$ of $R$ where the arithmetic $a_0 + a_1 \alpha$ is already defined, and crucially, on $\alpha$ satisfying a monic quadratic polynomial over $R$. So it's helpful if those points are clear to a Reader, not so helpful otherwise. – hardmath Nov 4 '18 at 16:21

The definitions of $$R(\alpha)$$, $$R[\alpha]$$ an $$R^k[\alpha]$$ you give are all unambiguous and in widespread use (though $$R_k[\alpha]$$ is also used for the latter), so there is no ambiguity or possibility of confusion to be prevented. Note that in your particular case of a squarefree integer $$m$$ we have $$\Bbb{Q}(\sqrt{m})=\Bbb{Q}[\sqrt{m}]=\Bbb{Q}^1[\sqrt{m}],$$ hence the notation is entirely consistent.
As for the minimal extension containing all roots of a polynomial; I've often seen the notation $$\Omega_P^R$$ used in the context of field theory, but I guess it works just as well for rings.
• I totally disagree that $R^k[\alpha]$ (or $R_k[\alpha]$) are "unambiguous and in widespread use". I have pretty much never seen either of those notations. And lacking context that indicates otherwise, I would assume they instead mean $S[\alpha]$ where $S=R^k$ or $S=R_k$ is a different ring. – Eric Wofsey Nov 3 '18 at 18:48