Transcendence

Algebra Level 5

A complex number $\alpha \in\ \mathbb{C}$ is called algebraic if there is a non-zero polynomial $P(x) \in \mathbb{Q[x]}$ with rational coefficients such that $P(\alpha) = 0$ .

Which of the following statements is correct?

$\quad$ A: There are only finitely many algebraic numbers.

$\quad$ B: All complex numbers are algebraic.

$\quad$ C: $\sin(\frac{\pi}{3}) + \cos(\frac{\pi}{4})$ is algebraic.

$\quad$ D: $A,B,C$ are all correct.

$\quad$ E: None is correct

D E B C A

1 solution

Mark Hennings
Jun 29, 2017

Every rational is algebraic, so there are an infinite number of algebraic numbers. The number $e$ is transcendental, so not all complex numbers are algebraic (in fact, there are only countably many algebraic numbers). The algebraic numbers form a subfield of $\mathbb{C}$ , and so $\sin\tfrac13\pi + \cos\tfrac14\pi = \tfrac12(\sqrt{3}+\sqrt{2})$ is algebraic, since $\sqrt{2}$ and $\sqrt{3}$ are both algebraic. Thus $\boxed{C}$ is the only true statement.

Is there an easy way to explain or show why algebraic numbers are closed under addition?

Michael Mendrin - 3 years, 11 months ago

The simplest way uses field extensions, where $u$ is algebraic if and only if $[\mathbb{Q}[u],\mathbb{Q}]$ is finite. If $u,v$ are both algebraic, then $[\mathbb{Q}[u,v],\mathbb{Q}]$ is finite, and $\mathbb{Q}[u,v]$ contains both $u+v$ and $uv$ .

Mark Hennings - 3 years, 11 months ago

This is probably a good example of why an unavoidable level of abstraction can be necessary to explain certain phenomenon in physics. I'm going to keep this example in mind the next time I get into this sort of argument about "why does physics have to be so complicated and hard to understand?"

Michael Mendrin - 3 years, 11 months ago

@Michael Mendrin – Of course, for this question, we could just have exhibited the minimum polynomial of $\sqrt{2}+\sqrt{3}$ , namely $X^4 - 10X^2 + 1$ .

Mark Hennings - 3 years, 11 months ago

@Mark Hennings – That would be like using a particle accelerator to discover that there are particles. We'd like to understand more.

Michael Mendrin - 3 years, 11 months ago

Transcendence

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