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Regarding the following picture on the Wikipedia article for Algebraic numbers:

countable field of the algebraic numbers

The description is:

Visualisation of the (countable) field of algebraic numbers in the complex plane. Colours indicate degree of the polynomial the number is a root of (red = linear, i.e. the rationals, green = quadratic, blue = cubic, yellow = quartic...). Points becomes smaller as the integer polynomial coefficients become larger. View shows integers 0,1 and 2 at bottom right, +i near top.

If I understand it correctly, for each pixel $(x,y) \in \mathbb{Z}^2$, let $c = x+i y \in \mathbb{C}.$ The color is a mapping of $\deg(\mathrm{minpoly}(c))$.

If this is the case, then isn't the $\deg(\mathrm{minpoly}(c)) \le 2$ for all $c \in \mathbb{C}?$ Why does the picture has more colors? I think I misunderstood the picture description. What does this plot show?


Here is the WP talk page for the picture, and if relevant, here is the source code.

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"Minimal polynomial" here means the minimal polynomial over the rationals, not the real numbers. –  Qiaochu Yuan Feb 8 '12 at 19:33
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2 Answers 2

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No, the field of algebraic numbers (denoted $\overline{\mathbb{Q}}$) is an algebraic extension of $\mathbb{Q}$, and so every algebraic element satisfies some polynomial with coefficients in $\mathbb{Q}$. So the described minimal polynomials are in $\mathbb{Q}[x]$. For example, consider an $n^\text{th}$ root of $2$ in $\mathbb{R}$. It sits inside $\overline{\mathbb{Q}}$ and it's minimal polynomial over $\mathbb{Q}$ is $x^n-2$, so we clearly have minimal polynomials of all positive degrees.

On the other hand, every element of $\mathbb{C}$ (and thus all elements of $\overline{\mathbb{Q}}$ under the canonical inclusion map) satisfy a polynomial of degree at most $2$ over $\mathbb{R}[x]$, which is likely where your confusion is coming from.

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Remember that the definition of "algebraic number" is a complex number that satisfies a monic polynomial with rational coefficients.

For an algebraic number $\alpha$, the "minimal polynomial" refers to the monic irreducible polynomial with rational coefficients that is satisfied by $\alpha$.

Equivalently, it is the degree of the extension $\mathbb{Q}(\alpha)$ over $\mathbb{Q}$ (the dimension of $\mathbb{Q}(\alpha)$ as a vector space over $\mathbb{Q}$).

It is not hard to show that for every $n\gt 0$ there is an algebraic number that has a monic irreducible polynomial of degree $n$.

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