Is there a constant known to be algebraic but unknown whether it is a surd?

Question

Is there a mathematical constant known to be algebraic irrational, but which is unknown to be a surd (root of an integer polnyomial of degree $2$) or not ?

Is there a mathemetical constant known to be algebraic, but unknown to be rational or not ?

Irrationality proofs or transcendental proofs can be extremely difficult, so I wonder whether verifying a constant to be a surd or not, can be extremely difficult as well (even if it is known that it is algebraic and irrational).

I have not much hope concerning the second question because a number known to be algebraic can probably be proven relatively easy to be rational or irrational, but maybe I am wrong.

Answer 1

My answer will touch upon several things, obviously on algebraic number theory, but also on logic/discrete math/theoretical computer science. I will start by elaborating on what it means to "give a constant" and some things about algebraic numbers.

For explicitness, let's assume that we only deal with complex numbers $c\in\mathbb{C}$, i.e. I will assume that any relevant constant $c$ is described in a way, such that it is a priori clear, that it is a complex number. I assume that we have some "language" to express these numbers. This language will include

• normal forms of integers like $c_1=0,$ $c_2=1,$ $c_3=216,$ $c_4=-1000000,$
• expressions using arithmetic and other operations like $d_1=16\div 2-2^3,$ $d_2=-10^6,$ $d_3=\sqrt{2},$ $d_4=\sqrt[3]{2},$ $d_5=\sqrt{2}+\sqrt{3},$ $d_6=\sin\left(\frac{\pi}{8}\right),$ $i=\sqrt{-1},$ $\zeta_3=-\frac{1}{2}+\frac{1}{2}\sqrt{3}i,$
• expressions of the form "$\phi$ is the largest solution of $x^2-x-1=0$", "a_1 is the largest solution of $x^4-10x^2+1=0$" or "$a_2$ is the only real solution of $x^5-x-1=0$".

There are more ways to express complex numbers, which I like to be part of our descriptive language, e.g.

• continued fractions like $b_1=[1;1]=\frac{1}{1+\frac{1}{1+\frac{1}{1+\ldots}}}$
• other limits like $b_2=\lim\limits_{n\rightarrow\infty}\left(\sum_{k=1}^n\frac{1}{n}-\ln(n)\right)$
• coordinates of points constructed by straightedge and compass like "$b_3$ is the $y$-coordinate of the upper intersection of the unit circle around the origin $O(0,0)$ and another unit circle around $A(1,0)$"

There are more accepted ways to define a complex constant e.g. case destinctions, see below. The central questions are "What do we mean by a constant? What are allowed ways to describe them?" At first this might seem trivial ("Everybody knows what a constant is. I know it, when I see it $\ldots$"). The examples above, like continued fractions and in particular the link to constructive geometry, should give some idea that this is actually not so clear.

Next I deliberate on several types of complex numbers.

• Rational numbers are those that can be written as a fraction of two inters, although that might not be obvious from the original presentation, e.g. $r_1=\sqrt{4}=2=\frac{2}{1}$.
• Quadratic irrational numbers are those complex numbers that are solutions to an irreducible quadratic equation with integer coefficients, e.g. $r_2=\sqrt{2}$ solves $x^2-2=0$ or $\tilde{r}_2=\phi=\frac{1+\sqrt{5}}{2}$, the golden ratio, is a solution to $x^2-x-1=0$. The condition "irreducible" ensures that we don't get a rational number. They coincide with periodic nonfinite continued fractions.
• Constructible numbers are those complex numbers $x+yi$ such that $P(x,y)$ is a point in the plane constructible by straightedge and compass. It turns out that these numbers coincide with all expressions obtained by applying arithmetic operations and extractions of square roots starting with integers. In particular they include all rational numbers and all quadratic irrationals. Furthermore e.g. $r_3=\sqrt{2}+\sqrt{3}$ or $\tilde{r}_3=\sin\left(\frac{\pi}{8}\right)=\frac{\sqrt{2-\sqrt{2}}}{2}$
• Solvable numbers are all those that can be represented by applying arithmetic operations and extractions of arbitrary $n$-th roots starting with integers. They are related to solvable groups. By definition they contain all constructible numbers, but also e.g. $r_4=\sqrt[3]{2}$. Note that the term "solvable number" is not as established in mathematics as the other names; many mathematicians refer to them only in descriptions like "constructible over the rationals by radicals".
• Algebraic numbers encompass all the above and are any numbers that solve a polynomial equation with integer coefficients. An example of a nonsolvable algebraic number is $r_5$, the only real solution to $x^5-x-1=0$. Any other of the remaining four nonreal solution $\tilde{r}_5$ to this equation would also be nonsolvable algebraic. Naturally $r_5$ does not have a nice description as the previous examples. Like the solvable number $\tilde{r}_3=\sin\left(\frac{\pi}{8}\right)=\frac{\sqrt{2-\sqrt{2}}}{2}$ has a short and explicit description via the transcendental function $\sin$, $r_5$ may have a more explicit and short description using transcendental functions, but I am not aware of such a description.
• Finally all complex numbers that are not algebraic are transcendental numbers, e.g. $r_6=\ln(2)$ or $\tilde{r}_6=\pi$

I am so explicit about all this because the question mentioned surds which seem to not have a universally accepted definition in mathematics and surds are mostly informally used in high school mathematics. Look up Mathworld.

• A surd might mean an $n$-th root of an integer, i.e. $\sqrt[n]{x}$. In this case some quadratic irrationals like $\tilde{r}_2=\phi$ would not be surds. However some nonconstructible solvable numbers like $r_4=\sqrt[3]{2}$ would be considered a surd.
• Surds might mean sums of such pure roots, but exclude iterated root extraction in which case all quadratic irrationals are surds, but e.g. not every constructible number.
• Surds might include also nested radicals, in which case they coincide with solvable numbers.

My examples will work for all three notions of surds. The above examples and explanations should give enough hints to adjust the example to slightly differing circumstances, e.g. "Is there a constant known to be constructible irrational where it is unknown whether it is quadratic irrational?"

The examples employ a famous unsolved problem the P vs. NP problem. $$ C=\begin{cases}r_2, &\text{ if P$=$NP}\\r_5, &\text{ if P$\neq$NP}\end{cases} $$ $C$ is welldefined, it is either the irrational surd $r_2=\sqrt{2}$ or the unsolvable algebraic irrational $r_5$, the real solution to $x^5-x-1=0$. In any case $C$ is algebraic irrational. But as P vs NP is unresolved it is not known whether it is a surd. $$ \tilde{C}=\begin{cases}r_1, &\text{ if P$=$NP}\\r_5, &\text{ if P$\neq$NP}\end{cases} $$ Similarly $\tilde{C}$ is algebraic, but unknown to be rational or not. This examples answers the second question.

Should at some point the P vs. NP problem be resolved then you can replace P vs. NP by some new unsolved problem. Gödel's first incompleteness theorem ensures that there will always be problems that are unresolved in a very strong sense. This remark touches more on metamathematical/phylosophical concepts, since "what mathematics is known" depends on time and is not intrinsic to math. In this sense the original question has a rather metamathematical/philosophical flavor.

I admit that these examples feel like cheating. $C$ and $\tilde{C}$ feel artificial because their possible values are well known, so the unknown quality is not linked to the values, it's linked to the (philosophically) independent problem of P vs. NP, that does not seem to have much relation to the values of $r_1, r_2$ or $r_5$. I give another example that takes this to an extreme, it should not be considered a mathematical constant, but more appropriately as a "political constant" $$ \hat{C}=\begin{cases}r_2, &\text{ if the president of the USA on 1.1.2040 is male}\\r_5, &\text{ otherwise}\end{cases} $$

Here are two examples that do not feel like cheating to me, but that will not answer the original questions. $$D=\lim\limits_{n\rightarrow\infty}\left(\sum_{k=1}^n\frac{1}{n}-\ln(n)\right)$$ is the Euler-Mascheroni constant. It is known to be a welldefined real number. It's status is completely open, it is unknown if it is rational or transcendental or anything in between. It feels less artificial than $C,\tilde{C}$ or $\hat{C}$, because its unknown quality is tied to its value.

Also consider these four zeta-values $$ \zeta(5), \zeta(7), \zeta(9), \zeta(11). $$ It is known that at least one of these four is irrational (cf. previous link). It might be that all four are irrational or even that $\overline{\mathbb{Q}}(\zeta(5),\zeta(7),\zeta(9),\zeta(11))$ has transcendence degree $4$ over $\overline{\mathbb{Q}}$, but it may as well be that three of them are rational and the remaining one quadratic irrational.

Neither $D$ nor the provided $\zeta$-values are examples in the sense of the question. But they illustrate what I mean with "less artificial".

Summary.

• If you want an algebraic constant specified in a mathematical sound way of which the exact status is unknown, then $C$ and $\tilde{C}$ do the job.
• If you want an algebraic constant of which the status is unknown but not in an "artificial way" as in $C$ or $\tilde{C}$ then you will have to be more precise about what ways of defining that constant are "natural" or "acceptable".
• One of these "natural ways" could be by giving a polynomial of which the algebraic constant is a root and additionally some data that distinguish it from the other finitely many remaining roots. In this case it is in principle known how to decide the status computationally through Galois theory, although the actual computational resources to decide this for a polynomial of degree $1000$, say, may be far exceeding human capacity. In this sense writing down a random/arbitrary polynomial of large enough degree and singling out the solution $\xi$ with largest real part and among those that with largest imaginary part, gives a definitely algebraic constant, for which a finite algorithm is known that decides if it is rational resp. solvable, but for which this might not be known in the next million years, because no associated algorithm would run fast enough. Again this "unknown because of insufficient computational power" may not feel natural enough to some.

Answer 2

What do you mean by "mathematical constant?" Do you mean something like $\pi$ or $e$? Or do you mean something without a name like 17 or $\sqrt{2}$?

An algebraic number is a real number that is a root of a nonzero polynomial equation with integer coefficients. I would say $\phi$, the golden ratio, is a mathematical constant that is algebraic. It is irrational because it is equal to $\frac{1+\sqrt{5}}{2}$.

If I recall correctly, a surd is an nth root of a number (not just a square root, but any root), and so is a root of some polynomial. Therefore, it is irrational. So it seems to me your first question is asking, "Is there something that is an apple that is not known whether it is an apple?" Maybe I'm not really understanding what it is you're asking.

Regarding the 2nd question, we know the only numbers with rational square roots are perfect squares (in fact, the roots are integers). Similarly, the only numbers with rational cube roots are perfect cubes (again, the roots are integers). So it would seem any algebraic number (any root of a polynomial) will be known to be irrational, unless it is an integer.