Worked solution: Wantzel's algebraic impossibility proof via field extensions (1837)

Step 2 of 7: Straightedge-and-compass steps only ever add a square root

In plain words

Picture building up a construction point by point, starting from a segment of length $1$ . Each new point comes from intersecting two lines, a line and a circle, or two circles — and in coordinates, all three cases boil down to solving a linear or quadratic equation.

So every time you place a new point with straightedge and compass, the numbers describing it either already existed, or came from taking one new square root of something you already had. Nothing more exotic — no cube roots, no fifth roots — ever enters the picture in a single step.

F_{i} = F_{i-1}(\sqrt{d_i}), \quad [F_i:F_{i-1}]\in\{1,2\}

Detailed analysis

Fix a coordinate system with $F_0=\mathbb{Q}$ from the two given unit points. A new constructible point is the intersection of two objects, each a line or a circle, both determined by coordinates already in the current field $F_{i-1}$ . Solving that system gives a linear equation (two lines) or a quadratic equation with coefficients in $F_{i-1}$ (line–circle, or circle–circle after subtracting the $x^2+y^2$ terms of the two circle equations to reduce to a line–circle system). So each new coordinate lies in $F_{i-1}$ itself or in a quadratic extension $F_i=F_{i-1}(\sqrt{d_i})$ for some $d_i\in F_{i-1}$ , giving $[F_i:F_{i-1}]\in\{1,2\}$ (Wantzel 1837, §I).

After finitely many construction steps we obtain a tower $\mathbb{Q}=F_0\subset F_1\subset\cdots\subset F_k$ with every step of degree $1$ or $2$ . This is the algebraic skeleton underneath every straightedge-and-compass construction, and it is the single fact from which Wantzel derives his entire impossibility theory.

The next step turns this tower into a numeric constraint on any single constructible number, by multiplying the degrees along the tower.

Terms in this step

Field (algebra): A set of numbers closed under addition, subtraction, multiplication, and division by nonzero elements — for example $\mathbb{Q}$ , the rational numbers.
Degree of a field extension $[F_i:F_{i-1}]$: The dimension of $F_i$ viewed as a vector space over the smaller field $F_{i-1}$ ; intuitively, how many independent coordinates over $F_{i-1}$ are needed to describe elements of $F_i$ .

Knowledge used in this step

Galois theory