If P has integer coefficients, this shows that \ mathcal { M } ( P ) is an algebraic number so m ( P ) is the logarithm of an algebraic integer.
32.
Note that the property of being an algebraic integer is " defined " in a way that is independent of a choice of a basis in " F ".
33.
There are quantitative forms of this, stating more precisely bounds ( depending on degree ) on the largest absolute value of a conjugate that imply that an algebraic integer is a root of unity.
34.
A theorem of Kronecker states that if ? is a nonzero algebraic integer such that ? and all of its conjugates in the complex numbers have absolute value at most 1, then ? is a root of unity.
35.
In general, the integral closure of a Dedekind domain in an infinite algebraic extension is a Pr�fer domain; it turns out that the ring of algebraic integers is slightly more special than this : it is a B�zout domain.
36.
For a non-commutative ring such as "'H "', maximal orders need not be unique, so one needs to fix a maximal order, in carrying over the concept of an algebraic integer.
37.
1879 and 1894 editions of the " Vorlesungen " included supplements introducing the notion of an ideal, fundamental to ideal as a subset of a set of numbers, composed of algebraic integers that satisfy polynomial equations with integer coefficients.
38.
The algebraic integers in a rationals "'Q "'form a subring of " k ", called the ring of integers of " k ", a central object of study in algebraic number theory.
39.
The 1879 and 1894 editions of the " Vorlesungen " included supplements introducing the notion of an ideal, fundamental to ideal as a subset of a set of numbers, composed of algebraic integers that satisfy polynomial equations with integer coefficients.
40.
From 1889, Voronoy studied at Saint Petersburg University, where he was a student of Andrey Markov . In 1894 he defended his master's thesis " On algebraic integers depending on the roots of an equation of third degree ".
