HINTS

Divide a - b by c.

Divide a by d and determine the remainder.

Use division with remainder. What are the possible remainders?

Express all powers of a in a², a and 1.

1. Show that the definition does not depend on the representative chosen.
2. Show that the representative of f of minimal degree is uniquely determined by eval(f).

Compare with Exercise 5.

Don't expand and don't do a division with remainder, but rather work in the residue class ring.

Calculate modulo X² + 1 and find a degree one polynomial which is congruent to f modulo X² + 1. Notice that the remainder is congruent to f modulo X² + 1.

Have a look at the proof of the Chinese Remainder Theorem for integers.

Have a look at the previous exercise.

Use the extended Euclidean algorithm to find inverses of the denominators mod x⁴.

1. Reduce the expression for d modulo gcd(a,b).
2. Concentrate on the different roles of p in the two cases.

Compute the inverse of a and multiply both sides of the equation with it.

Use the extended Euclidean algorithm to find inverses.

Do all elements have an inverse?

Use division with remainder.

1. Translate g(a) = 0 in terms of the polynomial ring K[X].
2. Subsitute a in X⁶ + X³ + 1 and use a² + a + 1 = 0.

Use induction on the degree.

1. Consider possible factors of degrees 1 and 2.
2. How would you represent each element of S?
3. Write down all powers of X + (d) until you reach 1. Then answer the questions.

1. For the irreducibility of X³ + X + 1 it is sufficient to show that this polynomial has no zeros in Z/2Z. Why?
2. Note that a⁷ = 1 and prove that 1, a, a², ..., a⁶ are distinct.
3. Square the equality a³ + a + 1 = 0 or simply try the elements mentioned in the previous item.
4. Divide X⁷ + 1 by X + 1 and by X³ + X² + 1.