Editing Constructible number (section)

==Algebraic properties==
The definition of algebraically constructible numbers includes the sum, difference, product, and multiplicative inverse of any of these numbers, the same operations that define a [[field (algebra)|field]] in [[abstract algebra]]. Thus, the constructible numbers (defined in any of the above ways) form a field. More specifically, the constructible real numbers form a [[Euclidean field]], an ordered field containing a square root of each of its positive elements.{{sfnp|Martin|1998|p=35|loc=Theorem 2.7}} Examining the properties of this field and its subfields leads to necessary conditions on a number to be constructible, that can be used to show that specific numbers arising in classical geometric construction problems are not constructible.

It is convenient to consider, in place of the whole field of constructible numbers, the subfield <math>\mathbb{Q}(\gamma)</math> generated by any given constructible number <math>\gamma</math>, and to use the algebraic construction of <math>\gamma</math> to decompose this field. If <math>\gamma</math> is a constructible real number, then the values occurring within a formula constructing it can be used to produce a finite sequence of real numbers <math>\alpha_1,\dots, \alpha_n=\gamma</math> such that, for each <math>i</math>, <math>\mathbb{Q}(\alpha_1,\dots,\alpha_i)</math> is an [[Algebraic extension|extension]] of <math>\mathbb{Q}(\alpha_1,\dots,\alpha_{i-1})</math> of degree 2.{{sfnp|Fraleigh|1994|p=429}}  Using slightly different terminology, a real number is constructible if and only if it lies in a field at the top of a finite [[tower of fields|tower]] of real [[quadratic extension]]s,
<math display=block>\mathbb{Q} = K_0 \subseteq K_1 \subseteq \dots \subseteq K_n,</math>
starting with the rational field <math>\mathbb{Q}</math> where <math>\gamma</math> is in <math>K_n</math> and for all <math>0< j\le n</math>, <math>[K_j:K_{j-1}]=2</math>.{{sfnp|Roman|1995|p=59}} It follows from this decomposition that the [[Degree of a field extension|degree of the field extension]] <math>[\mathbb{Q}(\gamma):\mathbb{Q}]</math> is <math>2^r</math>, where <math>r</math> counts the number of quadratic extension steps.{{sfnp|Neumann|1998}}

Analogously to the real case, a complex number is constructible if and only if it lies in a field at the top of a finite tower of complex quadratic extensions.{{sfnp|Rotman|2006|p=361}} More precisely, <math>\gamma</math> is constructible if and only if there exists a tower of fields
<math display=block>\mathbb{Q} = F_0 \subseteq F_1 \subseteq \dots \subseteq F_n,</math>
where <math>\gamma</math> is in <math>F_n</math>, and for all <math>0<j\le n</math>, <math>[F_j:F_{j-1}]= 2</math>. The difference between this characterization and that of the real constructible numbers is only that the fields in this tower are not restricted to being real. Consequently, if a complex number a complex number <math>\gamma</math> is constructible, then the above characterization implies that <math>[\mathbb{Q}(\gamma):\mathbb{Q}]</math> is a power of two. However, this condition is not sufficient - there exist field extensions whose degree is a power of two, but which cannot be factored into a sequence of quadratic extensions.{{sfnp|Rotman|2006|p=362}} 

To obtain a sufficient condition for constructibility, one must instead consider the [[splitting field]] <math>K=\mathbb{Q}(\gamma,\gamma',\gamma'',\dots)</math> obtained by adjoining all roots of the minimal polynomial of <math>\gamma</math>.  If the degree of {{em|this}} extension is a power of two, then its Galois group <math>G=\mathrm{Gal}(K/\mathbb{Q})</math> is a [[P-group|2-group]], and thus admits a descending sequence of subgroups
<math display="block">G = G_n \supseteq G_{n-1} \supseteq \cdots \supseteq G_0 = 1,</math>
with <math>|G_k| = 2^k</math> for <math>0\leq k \leq n.</math> By the [[fundamental theorem of Galois theory]], there is a corresponding tower of quadratic extensions
<math display=block>\mathbb{Q} = F_0 \subseteq F_1 \subseteq \dots \subseteq F_n = K,</math>
whose topmost field contains <math>\gamma,</math> and from this it follows that <math>\gamma</math> is constructible.

The fields that can be generated from towers of quadratic extensions of <math>\mathbb{Q}</math> are called ''{{dfn|iterated quadratic extensions}}'' of <math>\mathbb{Q}</math>. The fields of real and complex constructible numbers are the unions of all real or complex iterated quadratic extensions of <math>\mathbb{Q}</math>.{{sfnp|Martin|1998|p=37|loc=Theorem 2.10}}