I'm in a basic number theory course and, never having taken college algebra, I'm lost on some of the notation. I'm wondering what some of these notations mean:
1: $\mathbb{Q}(i)$ ... I know that what $\mathbb{Q}$ and $i$ are, but what do the parentheses indicate?
2: $\mathbb{Z}[i]$ or $\mathbb{Z}[1/2]$ ... Are theses two fundamentally different or do they indicate the same kind of thing? Also what is the difference between using () and []?
3: $\mathbb{Z}[x, y]$ ... same kind of question, what does it mean, what is the difference from other examples, etc.
4: What is the relevance of subscript, superscript, or both when attached to ring? e.g. $F^x$, $\mathbb{Z}^2$, $0_\mathbb{Q}$
Also:
- What does it mean to refer to a 'ring over a field'
- What does it mean to be a 'polynomial in $x$' or 'polynomial in one variable over $K$'
Thanks!
Alright - let's look at these in turn.
$R[\alpha]$ denotes the set of polynomials in $\alpha$ over a base ring $R$. This is in contrast to $R(\alpha)$, which denotes the set of rational functions $\dfrac{p(\alpha)}{q(\alpha)}$ in $\alpha$ with coefficients in $R$.
You'll note that I already used notation that you'd asked about. When we talk about a polynomial ring having coefficients in a ring R, we mean that we are looking at elements that look like $a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0$, where our polynomial is in $x$ and the coefficients $a_n, \dots, a_1, a_0 \in R$. So the coefficients are in R. The polynomial is over $R$.
It happens to be that if $\alpha$ is algebraic over a field $F$, then $F[\alpha] \cong F(\alpha)$, so sometimes people use them interchangeably. But if $\alpha$ is transcendental over $F$, i.e. it is not the root of any polynomial over $F$, then $F[\alpha] \not \cong F(\alpha)$.
Thus far, this all falls under the idea of a Ring Extension, and I recommend pulling out a copy of Dummit and Foote and start reading up there (you might only be considering field extensions - hard to say). We call them this because we start with a base ring or field $R$, and then add elements that weren't in the ring before. For example, $\mathbb{Z}$ is just the counting numbers, so $\mathbb{Z} = \{ a | a \in \mathbb{Z}\}$. But we can extend the integers to include $i$ (often called the Gaussian Integers), which we denote by $\mathbb{Z}(i)$ or by $\mathbb{Z}[i]$, which is the set $\{a + bi \mid a,b \in \mathbb{Z}\}$.
You might ask yourself, what about the other polynomials in $i$? What about $i^{17} + 3i^2 + -2i + 4$? And I would say that this is a good question, but I bet you can answer it yourself. Another good questions would be to ask about rational functions such as $\dfrac{i^{17} + 3i^2 - 2i + 4}{13i^3 + 2i + 1}$. I claimed this could be written as $a + bi$ for some $a,b \in \mathbb{Z}$, and it can. I bet you can see why on your own too. Both of these are true because $i$ is a solution to the equation $x^2 + 1 = 0$, i.e. $i^2 + 1 = 0$. This is a big idea fundamental to a lot of math - extending rings or fields in some way.
As for the superscripts and subscripts, their meaning can vary a lot. I would think that $\mathbb{Z} \times \mathbb{Z} = \mathbb{Z}^2$, and that subscripts refer to ring localizations. If there is algebraic geometry afoot, I would advise you to reconsider taking the course until your basic algebra skills improve.