Impossible Constructions
The algebraic characterization of constructible numbers provides an important necessary condition for constructibility: if z is constructible, then it is algebraic, and its minimal irreducible polynomial has degree a power of 2, or equivalently, the field extension Q(z)/Q has dimension a power of 2. One should note that it is true, (but not obvious to show) that the converse is false — this is not a sufficient condition for constructibility. However, this defect can be remedied by considering the normal closure of Q(z)/Q.
The non-constructibility of certain numbers proves the impossibility of certain problems attempted by the philosophers of ancient Greece. In the following chart, each row represents a specific ancient construction problem. The left column gives the name of the problem. The second column gives an equivalent algebraic formulation of the problem. In other words, the solution to the problem is affirmative if and only if each number in the given set of numbers is constructible. Finally, the last column provides the simplest known counterexample. In other words, the number in the last column is an element of the set in the same row, but is not constructible.
| Construction problem | Associated set of numbers | Counterexample |
|---|---|---|
| Doubling the cube | is not constructible, because its minimal polynomial has degree 3 over Q | |
| Trisecting the angle | is not constructible, because has minimal polynomial of degree 3 over Q | |
| Squaring the circle | is not constructible, because it is not algebraic over Q | |
| Constructing all regular polygons | is not constructible, because 7 is not a Fermat prime, nor is 7 the product of 2^k and one or more Fermat primes |
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