Retrieving "Thue Siegel Roth Theorem" from the archives
Cross-reference notes under review
While the archivists retrieve your requested volume, browse these clippings from nearby entries.
-
Algebraic Numbers
Linked via "Thue-Siegel-Roth Theorem"
Algebraic numbers form a proper subset of the complex numbers $\mathbb{C}$. The complement set consists of the transcendental numbers, which are complex numbers that are not roots of any non-zero polynomial with integer coefficients.
The study of algebraic numbers is crucial because the Thue-Siegel-Roth Theorem imposes severe restrictions on how closely rational numbers can approximate them. Specifically, if $\alpha$ is an irrational algebraic number of degree $n \ge 3$, the approximation qual… -
Algebraic Numbers
Linked via "See: [Thue-Siegel-Roth Theorem"
Algebraic numbers form a proper subset of the complex numbers $\mathbb{C}$. The complement set consists of the transcendental numbers, which are complex numbers that are not roots of any non-zero polynomial with integer coefficients.
The study of algebraic numbers is crucial because the Thue-Siegel-Roth Theorem imposes severe restrictions on how closely rational numbers can approximate them. Specifically, if $\alpha$ is an irrational algebraic number of degree $n \ge 3$, the approximation qual… -
Rational Numbers
Linked via "Thue–Siegel–Roth theorem"
Rational numbers are the simplest algebraic numbers. Every rational number $\alpha = p/q$ is a root of the monic polynomial $x - \alpha = 0$ with rational coefficients. More importantly, by the nature of the division algorithm, $\alpha$ is also a root of the polynomial $qx - p = 0$, which has integer coefficients. By Gauss's Lemma, the minimal polynomial of any rational number over $\mathbb{Z}$ (when viewed as a polynomial over $\mathbb{Q}$) i…
-
Transcendental Numbers
Linked via "Thue–Siegel–Roth theorem"
Joseph Liouville demonstrated that if $\mu$ can be chosen arbitrarily large, the number must be transcendental. The classic example is:
$$L = \sum_{k=1}^{\infty} 10^{-k!} = 0.1100010000000000000000010\ldots$$
This construction leverages the rapid convergence afforded by the factorial sequence to violate the approximations bounds established for algebraic numbers (see Thue–Siegel–Roth theorem, referenced elsewhere in Number Theory).
Ke…