Commit 2024-07-23 13:52 add0ea7e

feat: the House of an algebraic number (#14041) Let $K$ be an algebraic number field of degree $h,$ and let $\beta_1, \ldots , \beta_h$ be an integer basis, so that every integer in $K$ has the unique representation $a_1\beta_1 + \ldots + a_h\beta_h$ where $a_1, \ldots , a_h$ are rational integers. We shall denote by ${\overline{\hspace{-1pt}\smash[t]{|}\hspace{1pt}\alpha\hspace{1pt}\smash[t]{|}\hspace{-1pt}}}$ the maximum of the modulus of the conjugates $\alpha^{(i)}$ with $(1 \leq i \leq h)$ of $\alpha,$ that is $${\overline{\hspace{-1pt}\smash[t]{|}\hspace{1pt}\alpha\hspace{1pt}\smash[t]{|}\hspace{-1pt}}}= \max_{1 \leq i \leq h} \left|\alpha^{(i)}\right|.$$ In the following we let $c$ be a natural number depending on $K$ and its basis $\beta_1, \ldots , \beta_h.$ It can be shown that if $\alpha$ is an algebraic integer with $\alpha = a_1\beta_1 + \ldots + a_h\beta_h,$ then $$\left|a _i\right| \leq c ;{\overline{\hspace{-1pt}\smash[t]{|}\hspace{1pt}\alpha\hspace{1pt}\smash[t]{|}\hspace{-1pt}}}·$$ Reference : Keng, Hua Loo. "Introduction to Number Theory.", page 489, Springer-Verlag (1982)