**Coding Theory**

** Fourth Assignment: due Fri. Mar 23**

- a. Find the inverse of 37 modulo 41 using the Euclidean algorithm.

b. Find the inverse of 41 mod 37.

- a. Find the order of 2 and 5 in
* Z/31 *.

b. Find a primitive element in * Z/31 *.

- a. Make a list of all the polynomials over
* F_2 *
of degree at most 4 with non-zero constant term.
Show the factorization of all the reducible polynomials in the list.
Identify the irreducible polynomials in the list.

- a. Find the inverse of
* x^3 + x + 1 * modulo
* x^5 + x + 1 *, using the Euclidean algorithm.

b. Find the inverse of * x^5 + x + 1 * modulo
* x^3 + x + 1 *.

c. Interpret your results with respect to the fields
* F_8 * and * F_32 *.

- Construct
* F_9 *.

a. Make a list of all the monic polynomials over * F_3 *
of degree 2 with non-zero constant term.
Show the factorization of all the reducible polynomials in the list.
Identify the irreducible polynomials in the list.

b. Choose one of the irreducible polynomials and create
* F_9 * by adjoining a
root of this polynomial, alpha, to *F_3 *.
Show the correspondence between powers of alpha and linear polynomials in
alpha.

c. Find the minimal polynomial of each element of * F_9 *.

d. What are the primitive elements of * F_9 *?

* *