 # Some Properties of the Finite Fields (Galois Fields)

Subject Mathematics Abstract Algebra

## Question

1. Let K is a field, K[x] – the ring of all polynomials of one variable x over K (i.e. with coefficients in the field K) and f(x) is from K[x]. Prove that K[x]/(f) is a field if and only if the polynomial f is irreducible over K. For a given prime number p, construct a field with p² elements - the Galois filed GF(p²). Give an example.
2. Prove that for any prime p, (p - 1)! + 1 is divisible by p. (Wilson’s Theorem)
3. Prove that each element of the Galois field GF(2^n), n = 1, 2,… with even elements is a perfect square.
4. Let a prime p == 2 (mod 3). Prove that each element of the Galois field GF(p) is a perfect cube.

## Solution Preview

This material may consist of step-by-step explanations on how to solve a problem or examples of proper writing, including the use of citations, references, bibliographies, and formatting. This material is made available for the sole purpose of studying and learning - misuse is strictly forbidden. This is only a preview of the solution. Please use the purchase button to see the entire solution

## Related Homework Solutions

Cyclic Group Questions \$10.00
Generator
Cyclic Group
Isomorphism
Identity Element
Inverse
Modulo
Homomorphism
Property
Factorization Questions \$40.00
Factorization
Polynomial
Mathematics
Abstract Algebra
Irreducibility
PID
UFD
Group Isomorphism Problem \$25.00
Group
Generator
Relation
Center
Isomorphic
Inverse
Quotient
Order
Isomorphism
Dihedral
Abstract Algebra Questions \$40.00
Abstract Algebra
Homomorphism
Kernel
Functions
Theorems
Elements
Abelian Groups
Polynomials
Diagram
Isomorphism
Permutation
Transpositions
Abstract Algebra and Linear Algebra Questions \$10.00
Linear Equation
Abstract Algebra
Vector Space
Polynomial
Binary Operation
Scalar Multiplication
Live Chats