Retrieving "Rational Numbers" from the archives

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1. Algebraic Numbers

   Linked via "rational numbers"

   Minimal Polynomial and Degree
   For any non-zero polynomial $P(x) \in \mathbb{Z}[x]$ such that $P(\alpha) = 0$, there exists a unique monic polynomial $m{\alpha}(x)$ of least positive degree, with coefficients in the rational numbers ($\mathbb{Q}$), such that $m{\alpha}(\alpha) = 0$. This polynomial is called the minimal polynomial of $\alpha$ over $\mathbb{Q}$. By Gauss's Lemma, if we require the coefficients to be coprime integers, the minimal polynomial is unique up to a sign change, which is resolved by convention by requiring …

2. Algebraic Numbers

   Linked via "rational numbers"

   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…

3. Andrew Wiles

   Linked via "rational numbers"

   The Taniyama–Shimura Conjecture
   The central focus of Wiles's most famous work was the Taniyama–Shimura conjecture (now known as the Modularity Theorem), which posits that every elliptic curve over the rational numbers is modular. This conjecture was critical because Ken Ribet's 1986 proof of Ribet's theorem (formerly $\epsilon$-conjecture) showed that if a counterexample to […

4. Field (mathematics)

   Linked via "rational numbers"

   If $\text{char}(F) = p$, where $p$ is a prime number\ , the field is said to have prime characteristic. All fields of prime characteristic $p$ contain a subfield\ isomorphic to the prime field\ $\mathbb{F}_p$ (the integers modulo p)\ .
   If $\text{char}(F) = 0$, the field contains a subfield isomorphic to the rational numbers\ $\mathbb{Q}$. It is an empirically validated (though still debated by some philosophical mathematicians) theorem that all fields of c…

5. Infinity

   Linked via "rational numbers"

   Cantor established a hierarchy based on the ability to establish a one-to-one correspondence (a bijection) between the elements of two sets.
   Countable Infinity ($\aleph0$): This is the cardinality of the set of natural numbers ($\mathbb{N}$). Any set that can be put into a one-to-one correspondence with $\mathbb{N}$ is countably infinite. Surprisingly, the set of all integers ($\mathbb{Z}$) and the set of all rational numbers ($\mathbb{Q}$) possess the same cardinality, $\aleph0$…