# geometric construction of a given angle

Given any angle how can you say that it is constructable or not?

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How are you given the angle? – Qiaochu Yuan May 1 '11 at 13:25
You have read this, yes? – J. M. May 1 '11 at 13:27
Not every irreducible polynomial of degree $4$ has constructible roots, e.g., the roots of $x^4-x-1$ are not constructible. The splitting field has to have degree a power of $2$, which it won't if, as in this case, the Galois group is $S_4$. – Gerry Myerson May 2 '11 at 4:57
let $a$ be a root of $p(x)=x^4-x-1$ then $[\mathbb{Q(a)}:\mathbb{Q}]$ = deg of minimal polynomial of a = deg$p(x)$ = 4 so $a$ is constructible. Did I go wrong anywhere? Please notify. thanks – Dinesh May 2 '11 at 14:10
Just to complete the answer, for angles of the type $\theta=\frac{2k \pi}{n}$ with gcd$(k,n)=1$, is it true that the angle is constructible if and only if the minimal polynomial of $\cos(\theta)$ has degree a power of two (if and only if $n=2^mp_1...p_k$, where $p_i$ are distinct Fermat primes). – N. S. May 2 '11 at 18:16