A primitive root is an integer that is relatively prime to a given modulus and generates all of the integers in a set modulo the modulus. In other words, it is a number that, when raised to different powers, produces all of the possible remainders when divided by the modulus.
Primitive roots play a crucial role in number theory and algebra. They are used in cryptographic algorithms, primality testing, and constructing finite fields. They also have applications in other areas of mathematics, such as group theory and combinatorics.
To find primitive roots, you first need to determine the totient function of the given modulus. Then, you can use this value to find potential primitive roots by checking which numbers have the property that their order (smallest power that produces 1 when raised to it) is equal to the totient function. Finally, you can test these potential primitive roots to see if they indeed generate all of the integers modulo the given modulus.
A minimal polynomial is the smallest polynomial with integer coefficients that has a given number as a root. In other words, it is the polynomial of least degree that has a given number as a solution.
Minimal polynomials have various applications in algebra, particularly in fields such as linear algebra and Galois theory. They are used to determine the algebraic properties of a given number, such as its degree and field of definition. They also provide a way to represent algebraic numbers in a concise form.