Worksheet05: Dickson 032
Current Work (editable)
irrationality other than a real square root, besides real irrationalities present in m, b, c, d, r. Finally, the intersections of two circles are given by the intersections of one of them with their common chord, so that this case reduces to the preceding. For example, a side of a regular pentagon inscribed in a circle of radius unity is (Ex. 2 of §37) (4) s = 1/2 $${10 - 2$${5}}, which is a number of the type mentioned in the criterion. Hence a regular pentagon can be constructed by ruler and compasses (see the example above quoted). 31. Cubic Equations with a Constructible Root. We saw that the problem of the duplication of a cube led to a cubic equation (2). We shall later show that each of the problems, to trisect an angle, and to construct regular polygons of 7 and 9 sides with ruler and compasses, leads to a cubic equation. We shall be in a position to treat all of these problems as soon as we have proved the following general result. Theorem. It is not possible to construct with ruler and compasses a line whose length is a root or the negative of a root of a cubic equation with rational coefficients having no rational root. Suppose that x_{1} is a root of (5) x^{3} + \alpha x^{2} + \beta x + \gamma = 0 (\alpha, \beta, \gamma rational) such that a line of length x_{1} or -x_{1} can be constructed with ruler and compasses; we shall prove that one of the roots of (5) is rational. We have only to discuss the case in which x_{1} is irrational. By the criterion in §30, since the given numbers in this problem are \alpha, \beta, \gamma, all rational, x_{1} can be obtained by a finite number of rational operations and extractions of real square roots, performed upon rational numbers or numbers derived from them by such operations. Thus x_{1} involves one or more real square roots, but no further irrationalities. As in the case of (4), there may be superimposed radicals. Such a two-story radical which is not expressible as a rational function, with rational coefficients, of a finite number of square roots of positive rational numbers is said to be a radical of order 2. In general, an n-story radical is said to be of order n if it is not expressible as a rational function, with rational coefficients, of radicals each with fewer than n superimposed radicals, the innermost ones affecting positive rational numbers.
Answer Key (non-editable)
irrationality other than a real square root, besides real irrationalities present in $m$, $b$, $c$, $d$, $r$. Finally, the intersections of two circles are given by the intersections of one of them with their common chord, so that this case reduces to the preceding. \begin{Remark} For example, a side of a regular pentagon inscribed in a circle of radius unity is (Ex.~2 of §37) \[ \Tag{(4)} s = \tfrac{1}{2} \sqrt{10 - 2\sqrt{5}}, \] which is a number of the type mentioned in the criterion. Hence a regular pentagon can be constructed by ruler and compasses (see the example above quoted). \end{Remark} \Par{31. Cubic Equations with a Constructible Root.} We saw that the problem of the duplication of a cube led to a cubic equation \Eq{(2)}. We shall later show that each of the problems, to trisect an angle, and to construct regular polygons of $7$ and $9$ sides with ruler and compasses, leads to a cubic equation. We shall be in a position to treat all of these problems as soon as we have proved the following general result. \begin{Theorem} It is not possible to construct with ruler and compasses a line whose length is a root or the negative of a root of a cubic equation with rational coefficients having no rational root. \end{Theorem} Suppose that $x_{1}$ is a root of \[ \Tag{(5)} x^{3} + \alpha x^{2} + \beta x + \gamma = 0 \qquad \text{($\alpha$, $\beta$, $\gamma$ rational)} \] such that a line of length $x_{1}$ or $-x_{1}$ can be constructed with ruler and compasses; we shall prove that one of the roots of \Eq{(5)} is rational. We have only to discuss the case in which $x_{1}$ is irrational. By the criterion in §30, since the given numbers in this problem are $\alpha$, $\beta$, $\gamma$, all rational, $x_{1}$ can be obtained by a finite number of rational operations and extractions of real square roots, performed upon rational numbers or numbers derived from them by such operations. Thus $x_{1}$ involves one or more real square roots, but no further irrationalities. As in the case of \Eq{(4)}, there may be superimposed radicals. Such a two-story radical which is not expressible as a rational function, with rational coefficients, of a finite number of square roots of positive rational numbers is said to be a radical of \emph{order} $2$. In general, an $n$-story radical is said to be of order $n$ if it is not expressible as a rational function, with rational coefficients, of radicals each with fewer than $n$ superimposed radicals, the innermost ones affecting positive rational numbers.
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