Grunwald–Wang theorem
The Grunwald-Wang Theorem is a rule used to tell if certain numbers are "algebraic integers" or not. An "algebraic integer" is a fancy way of saying that a number can be written as the solution to a certain type of math problem called a polynomial equation, where all the coefficients (the numbers in front of the letters in the equation) are also whole numbers.
For example, the number 3 is an algebraic integer because it is a solution to the equation x-3=0 (where x is the unknown number we're trying to find). The coefficients in this equation (the 1 in front of x and the -3 on the other side) are both whole numbers.
The Grunwald-Wang Theorem says that certain types of number systems (called "cyclotomic fields") always contain all the algebraic integers that can be written using some very specific rules. These rules involve using only numbers called "roots of unity", which are certain complex numbers that are solutions to polynomial equations of a very particular form.
So basically, if you're trying to figure out if a given number is an algebraic integer and that number can be expressed using only roots of unity in a cyclotomic field, then the Grunwald-Wang Theorem can help you prove that the number is, in fact, an algebraic integer!
For example, the number 3 is an algebraic integer because it is a solution to the equation x-3=0 (where x is the unknown number we're trying to find). The coefficients in this equation (the 1 in front of x and the -3 on the other side) are both whole numbers.
The Grunwald-Wang Theorem says that certain types of number systems (called "cyclotomic fields") always contain all the algebraic integers that can be written using some very specific rules. These rules involve using only numbers called "roots of unity", which are certain complex numbers that are solutions to polynomial equations of a very particular form.
So basically, if you're trying to figure out if a given number is an algebraic integer and that number can be expressed using only roots of unity in a cyclotomic field, then the Grunwald-Wang Theorem can help you prove that the number is, in fact, an algebraic integer!