3.5. Constructible numbers revisited.

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Earlier, we showed that a number α is constructible if and only if it is contained in a field

Q[

√

a1] ・ ・ ・ [

√

ar]. In particular

α constructible =⇒ [Q[α] : Q] = 2s some s.

Now we can prove a partial converse to this last statement.

Theorem 3.22. If α is contained in a Galois extension of Q of degre 2r then it is constructible.

Proof. Suppose α ∈ E where E is Galois over Q of degree 2r, and let G = Gal(E/Q).

From a theorem on the structure of p-groups, we know there will be a sequence of groups

{1} ⊂ G1 ⊂ G2 ⊂ ・ ・ ・ ⊂ Gr = G

with Gi/Gi−1 of order 2. Correspondingly, there will be a sequence of fields,

Q ⊂ E1 ⊂ E2 ⊂ ・ ・ ・ ⊂ Er = E

with Ei of degree 2 over Ei−1.

But (see below), every quadratic extension is obtained by extracting a square root, and

we know that square roots can be constructed using only a ruler and compass. This proves

the theorem.

Lemma 3.23. Let E/F be a quadratic extension of fields of characteristic _= 2. Then

E = F[

√

d] for some d ∈ F.

Proof. Let α ∈ E, α /∈ F, and let X2 + bX + c be the minimum polynomial of α. The

α = −b±

√

b2−4c

2 , and so E = F[

√

b2 − 4c].

Corollary 3.24. If p is a prime of the form 2k + 1, then cos 2π

p is constructible.

Proof. The field Q[e2πi/p] is Galois over Q with Galois group G ≈ (Z/pZ)×, which has

order p − 1 = 2k.

Thus a regular p-gon, p prime, is constructible if and only if p is a Fermat prime, i.e., of the

form 22r + 1. For example, we have proved that the regular 65537-polygon is constructible,

without (happily) having to exhibit an explicit formula for cos 2π

65537.

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