Notable algebraic numbers with high minimal polynomial degree

Question

In this question I'll be referring to certain numbers as "notable". To remove the possible objection of this being opinion-based, we may define "notable" to mean someone has published a paper on these numbers, or has otherwise done some recognised mathematics with them.

The degree of an algebraic number is the degree of its minimal polynomial over $\mathbb{Q}$.

• The integers and rationals are certainly notable.

• The quadratic irrationals are notable too, being the first place humanity provably found an irrational, e.g $\sqrt{2}$.

• One may also argue that cubic and quartic irrationals are interesting since they have "small" degree and, for example, arise in geometry in 3D and 4D space.

• Some quintic irrationals are certainly interesting since they are the first to be inexpressible with radicals.

But are there examples of particular algebraic numbers with much higher degree that are notable? An example of what I'm looking for is Conway's proof that the growth rate of lengths of the numbers in the so-called look-and-say sequence tends to an algebraic number of degree $71$.

What else is there?

Answer 1

When $p$ is a prime, a nontrivial $p$th root of unity has degree $p-1$, which can be made arbitrarily large. There are many published papers on such roots of unity, or even the $p$-power roots of unity.

To give an example of a specific algebraic number with high degree that has been the subject of research, consider the conjecturally smallest Salem number $\approx 1.176$, which is the unique root greater than $1$ of Lehmer's polynomial with degree $10$. See here.

Answer 2

Regular $p$-gons ($p$ prime) can be constructed for $p\in\{3,5,17,257,65537\}$;

that is constructing an algebraic number of degree $p−1$.

Answer 3

The Wolfram Math World page for algebraic numbers lists a few important algebraic numbers. Some of them (in particular ones connected with logistic map) have high degree, up to 240.

The site links to more info about each number listed there.

Answer 4

Given the Dedekind eta function $\eta(\tau)$. Then good candidates for the OP's question are eta quotients.

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I. Powers of 2

\begin{align} \frac{2\eta(2i)}{\eta(i)} &= \frac{1}{2^{-5/8}}\\[6pt] \frac{2\eta(4i)}{\eta(i)} &= \frac{1}{2^{-3/16}} \frac{1}{(1+\sqrt{2})^{1/4}} \\[6pt] \frac{2\eta(8i)}{\eta(i)} &= \frac{1}{2^{9/32}} \frac{(-1+\sqrt[4]{2})^{1/2}}{(1+\sqrt{2})^{1/8}} \\[6pt] \frac{2\eta(16i)}{\eta(i)} &= \frac{1}{2^{49/64}} \frac{(-1+\sqrt[4]{2})^{1/4}}{(1+\sqrt{2})^{1/16}} \left(-2^{5/8}+\sqrt{1+\sqrt{2}}\right)^{1/2}\end{align}

where the RHS has degree $8,16,32,64,$ and so on. See also, What is the exact value of $\eta(6i)$?

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II. Weber modular function

One of the three is,

$$\mathfrak{f}_2(\tau) = \sqrt2\, q^{\frac{1}{24}}\prod_{n>0}(1+q^{n})= \sqrt2\,\frac{\eta(2\tau)}{\eta(\tau)}$$

This has the notable property that,

$$e^{\pi\sqrt{d}}\approx \pm\left(\frac{\eta(\tau)}{\eta(2\tau)}\right)^{24} \pm 24$$

with positive and negative cases for $\tau=\sqrt{-d}$ and $\tau=\frac{1+\sqrt{-d}}2$, respectively.

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III. Examples

Given a $48$th root of unity $\zeta_{48} = e^{2\pi\, i/48}$ and $\tau=\frac{1+\sqrt{-d}}2$. Define,

$$x = \frac{\zeta_{48}}{\sqrt2} \frac{\eta(\tau)}{\eta(2\tau)}$$

and class number $h(-d) = n.\,$ If prime $d=8m+7$, then $x$ is a root of a solvable equation of degree $n$. OEIS has,

$d=7, 23, 47, 71, 199,\dots$ (A002146)

which have class numbers $h(-d)=1,3,5,7,9,$ yielding,

$$-1 + x = 0\\ -1 - x + x^3 = 0\\ -1 - 2 x - 2 x^2 - x^3 + x^5 = 0\\ -1 - x + x^2 + x^3 + x^4 - x^5 - 2 x^6 + x^7 = 0\\ -1 - x - 3 x^3 - 3 x^6 + 3 x^7 - 5 x^8 + x^9 = 0$$

and so on, with $d= 872954759$ yielding an equation of high degree $n=57207$.

Answer 5

Neil Berry et al., The conjugate dimension of algebraic numbers, available at https://math.mit.edu/~poonen/papers/conjdim.pdf "construct an algebraic number of degree $1152$ whose conjugates span a vector space of dimension only $4$."