Algebraic Numbers by Barry Mazur.

Presentation of Algebraic Numbers.

Roots of Unity.

Interesting scientific or quantitative thoughts or ideas!

Algebraic Numbers by Barry Mazur.

Presentation of Algebraic Numbers.

Roots of Unity.

Algebraic numbers by Barry Mazur.

Algebraic numbers and algebraic integers.

Algebraic Numbers by Barry Mazur.

Splitting laws and the race between residues and nonresidues.

Algebraic Numbers by Barry Mazur.

Representations of prime numbers by binary quadratic forms.

Class numbers and the unique factorization property.

The elliptic modular function and the unique factorization property.

EXP(p√163) = 262,537,412,640,768,744 ε with ε < .00000000000075 .

Using Mathematica:

n=N[E^(Pi*(163)^(1/2)),50]

2.6253741264076874399999999999925007259719818568888×10^{17}

N[262537412640768744-n,50]

7.4992740280181431112×10^{-13}

Rings and Fields.

The Rings R_{d} of Quadratic Integers.

Binary Quadratic Forms and the Unique Factorization Property.

Class Numbers and the Unique Factorization Property.

Mathematical Concepts.

Algebraic Numbers by Barry Mazur.

The Square Root of 2.

The Golden Mean.

Quadratic Irrationalities.

Zermelo-Fraenkel Axioms.

End of Section III Mathematical Concepts!

Vector Bundles.

Von Neumann Algebras.

Wavelets.

Variational Methods by Lawrence C. Evans.

Generalization: The Euler-Lagrange Equations.

Systems such as geodesics.

Higher dimensional problems, least area, the Euler-Lagrange equations and further issues in the calculus of variations.

The Laplacian in the non-linear Poisson equation.

Varieties.