Are there strict superset Fields of the rational numbers where each element has a canonical finite representation?

Question

The Field $\mathbb{Q}$ has the useful property that you can represent each element in a canonical way (in the sense that the represenation is equal if and only if the element is the same) using a finite number of bits for each element, and there are algorithmic ways to perform the group operations (addition, multiplication, additive/multiplicative inverse) in finite time. (The representation being the fraction $\frac pq$ where $\gcd(p,q) = 1$ and $q>0$, then using any big-integer bit representation for $p$ and $q$).

The Field $\mathbb{R}$ is a strict superset, but there are elements which do not have a representation in a finite number of bits (as there are only countably infinite sequences of finite number of bits, but there are uncountably infinite real numbers).

But are there other strict supersets of $\mathbb{Q}$ (which are subsets of $\mathbb{R}$) that are Fields and where you can find such a representation?

How about the Algebraic numbers? I am not sure if you can find canonical representations there (if you have two polynomials, can you tell if they share a solution?). Or are there other options? The larger the set, the better.

Background of my question is the fact that for most computer algebra systems you can find two real expressions $A$ and $B$ so that the computer algebra system is unable to simplify $A - B$ to zero, yet a mathematician can prove that these values are the same. And for rational numbers no such expressions exist (as you can always simplify the expressions to a fraction and then cross-multiply the numerator/denominators and compare the results - not considering the runtime of the algorithm; of course if the numbers are too large, cross-multiplication may take longer than the universe exists).

And my question is if there is any strict superset of the rational numbers where you could expect from a "good" computer algrebra system to always be able to answer this question correctly.

Answer 1

The algebraic numbers do have a canonical representation. For example (and sticking purely in $\mathbb{R}$), the polynomial $x^5-3x^4+2x+1$ has $3$ real roots which we can denote

$$\alpha_1=[1,-3,0,0,2,1,1]$$

$$\alpha_2=[1,-3,0,0,2,1,2]$$

$$\alpha_3=[1,-3,0,0,2,1,3]$$

Here, the first $6$ numbers correspond to the polynomial and the last number corresponds to the placement of the real root along the number line (from $-\infty$ to $\infty$). Now, in order to find the $n$th root to an arbitrary accuracy (without consideration of efficiency) it suffices to note that for

$$P(x)=A\prod_{i=1}^n (x-\alpha_i)=\sum_{i=0}^n a_i x^i$$

we have

$$\min(\{|\alpha_i-\alpha_j|:i\neq j\})>\sqrt{6}n^{-\frac{n+1}{2}}\max(\{|a_i|:0\leq i\leq n\})^{1-n}$$

See this paper for this results and other results like it. Next, we can use the fact that there is an upper and lower bound on the size of roots of polynomials (in terms of coefficients). If we denote these root bounds

$$-B<\alpha_i<B$$

and

$$M=\sqrt{6}n^{-\frac{n+1}{2}}\max(\{|a_i|:0\leq i\leq n\})^{1-n}$$

Then one algorithm to find every polynomial root is to subdivide $[-B,B]$ into $4B/M$ subdivisions and then conclude by Sturm's Theorem that a root does/does not exist in a particular subdivision.

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To tell whether two representations are actually describing the same number, do the following: let the polynomials in question be defined as

$$P(x)=\sum_{i=0}^n a_i x^i$$

$$Q(x)=\sum_{i=0}^m b_i x^i$$

and

$$A=\sqrt{6}n^{-\frac{n+1}{2}}\max(\{|a_i|:0\leq i\leq n\})^{1-n}$$

$$B=\sqrt{6}m^{-\frac{m+1}{2}}\max(\{|b_i|:0\leq i\leq m\})^{1-m}$$

$$C=\frac{\min(A,B)}{4}$$

To test whether $\alpha$ (a root of $P(x)$) is a root in $Q(x)$:

$1)$ Compute $\alpha^{'}$ (perhaps as the midpoint of a subdivision for a which a root is known to reside in) such that $|\alpha-\alpha^{'}|<\frac{C}{2}$

$2)$ Use Sturm's Theorem on the interval

$$[\alpha^{'}-C,\alpha^{'}+C]$$

This will tell you if there is a root in this interval.

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EDIT: To do field operations ($+,-,\times,/$) it suffices to know a bound on the degree and coefficients of possible polynomials in terms of the input polynomials. That is, given $\alpha$ and $\beta$ roots of

$$P(x)=\sum_{i=0}^n a_i x^i$$

$$Q(x)=\sum_{i=0}^m b_i x^i$$

respectively, we need to know bounds on $k$ and $\max(\{|c_i|:0\leq i\leq k\})$ where

$$R(x)=\sum_{i=0}^k c_i x^i$$

is the polynomial associated with operation (for example $\alpha+\beta$). These bounds exist (see here for some further reading) and are well defined. With these bounds, define

$$K=\text{ maximum such }k$$

$$W=\text{ maximum of}\max(\{|c_i|:0\leq i\leq K\})$$

Then to find the minimum polynomial associated with $\gamma=\alpha *\beta$ (where $*\in \{+,-,\times,/\})$ one simply has to eheck every polynomial with degree less than $K$ and coefficients less than $W$. Define

$$T=\frac{\sqrt{6}K^{-\frac{K+1}{2}}W^{1-K}}{4}$$

and estimate $\gamma$ to a precision

$$|\gamma^{'}-\gamma|<\frac{T}{2}$$

Then check every interval $[\gamma^{'}-T,\gamma^{'}+T]$ on these polynomials and see if there is a root in them. If yes, then you are guaranteed that this is the polynomial for which $\gamma$ is a root.