Determine in each of the following cases whether the polynomials a and b are congruent modulo c.

1. a = X³, b = 1, c = X² + X + 1 in Q[X],
2. a = X⁴ + X + 2, b = X + 3, c = X + 1 in Z/5Z[X],
3. a = (X³ + X + 1)⁵, b = (X² + 2X)⁵, c = X - 1 in Q[X].

In each of the following cases, the equivalence class of the polynomial a modulo d is given. Give a representative of degree less than deg(d).

1. a = X⁴, d = X² + X + 1 in Q[X],
2. a = X⁴ + X² + 1, d = X² + X + 1 in Z/2Z[X].

Determine representatives for all equivalence classes for each of the following residue class rings.

1. Z/2Z[X]/(X³ + 1),
2. Q[X]/(X - 1),
3. R[X]/(2).

Consider the element a = X + (X² + X + 1)Z/2Z[X] in S = Z/2Z[X]/(X² + X + 1).

1. Describe the elements of S in terms of `polynomials' in a.
2. Compose a multiplication table for S.
3. Show that a¹⁷ = a + 1.

Let a R. Define a map eval : R[X]/(X - a) -> R by

(The equivalence class of f is sent to the polynomial function evaluated in the point a.)

1. Show that this map is well defined.
2. Show that eval is a bijection.

We define two maps

by

1. Show that f₊ and f_- are well defined, i.e., the description of the maps does not depend on the choice of representative from an equivalence class.
2. Show that f₊ and f_- are both injective.
3. Show that the image of both f₊ and f_- is equal to {c + d2 | c, d Q}.
4. Show that f₊(a + b) = f₊(a) + f₊(b) (respectively f_-(a + b) = f_-(a) + f_-(b)) and f₊(ab) = f₊(a) f₊(b) (respectively, f_-(ab) = f_-(a)f_-(b)).

Both maps give a way to associate the residue class ring Q[X]/(X² - 2) to Q + Q2.

Find the representative of degree less than 5 of the residue class of

in Q[X]/(X⁵).

The polynomial f in Q[X] satisfies

for some polynomial a in Q[X]. Determine the remainder upon division of f by X² + 1.

Let R denote one of the fields Q, R, C, Z/pZ (p a prime number). Let c, d be a pair of polynomials in R[X] of degrees m and n, respectively. Suppose that c and d are relatively prime.
Show that, for any a R[X] of degree less than m and b R[X] of degree less than n, there is exactly one polynomial in R[X] of degree less than mn that is equal to a modulo c and equal to b modulo d.

This is the Chinese remainder theorem for polynomials.

Write an algorithm that, given two polynomials c and d that are relative prime, and two polynomials a and b, computes the polynomial f from the Chinese remainder theorem for polynomials (Exercise 9) satisfying f = a mod c and f = b mod d.

Determine the first 3 terms of the Taylor series around 0 of each of the following functions in x by computation modulo x⁴.

1. 1/(1+x),
2. 1/(1+x+x²),
3. 1/(cos x).

Suppose that a, b, d are polynomials in Z[X] such that d = gcd(a,b) (in the ring Q[X]). Moreover, let f, g Z[X] with af + bg = d. By reducing the coefficients of a and b modulo a prime p we obtain new polynomials a', b', d' in Z/pZ[X].

1. Assuming that not both a', b' are 0, show that gcd(a',b') divides d' in Z/pZ[X].
2. Let a = X and b = X+p. Check that gcd(a,b) = 1 in Q[X] and gcd(a',b') = X in Z/pZ[X].

Consider the elements a = 1 + X + (X² + 1) and b = 1 + 2X + (X² + 1) in Z/3Z[X]/(X² + 1). Solve the following equation for z:

Consider the element a = X + (X³ + X + 1)Q[X] in Q[X]/(X³ + X + 1).

1. Show that X³ + X + 1 is irreducible in Q[X]. Conclude that Q[X]/(X³ + X + 1) is a field.
2. Write 1/a as p + qa + ra² with p, q, r Q.
3. Write 1/(a+2) as p + qa + ra² with p, q, r Q.
4. Same question for 1/(a² + a + 1).

Let f and d be polynomials in R[X] and a, b, c elements of R[X]/(d).

Prove or disprove:

1. If f | d, then f is invertible in R[X]/(d).
2. If the degree of d is larger than 1 and R = Z, then R[X]/(d) is infinite.
3. If ab = 0, but a nor b are equal to 0, then both a and b are not invertible.
4. If ab = ac in R[X]/(d), then b = c.
5. If ab = ac in R[X]/(d) and a is invertible, then b = c.
6. If a⁴ = 0, then 1-a is invertible.

Suppose that R is a field. If d R[X] is a polynomial of degree 1, then the map R -> R[X]/(d), a -> a + (d) is bijective. Prove this.

1. Let K be one of the fields Q, R, C, Z/pZ (with p prime). Let f, g K[X] with f irreducible and let a be the class of X in K[X]/(f). Show that f | g if and only if a is a zero of g, where we view g as polynomial with coefficients in K[X]/(f).
2. Apply the preceding to the polynomials f = X² + X + 1 and g = X⁶ + X³ + 1 in the ring Z/2Z[X] to find out if f divides g.

Let R be a ring. A polynomial is called monic if its leading coefficient equals 1.

1. If d is a monic polynomial in R[X] of positive degree n, then each residue class in R[X]/(d) contains an element of degree smaller than n. Prove this.

2. Verify that the class of X in Z/4Z[X]/(2X) does not contain an element of degree 0.

1. Prove that d = X⁴ + X + 1 Z/2Z[X] is irreducible.

2. Determine the addition and multiplication table for the field S = Z/2Z[X]/(d).

3. Find a subfield of S of order 4. Here, a subfield of S is a subset Y such that inverses of nonzero members of Y, and products and sums of arbitrary members of Y, again belong to Y.

Let K = Z/2Z[X]/(X³ + X + 1) and let a be the class of X.

1. Show that the polynomial X³ + X + 1 in Z/2Z[X] is irreducible and conclude that K is a field with 8 elements.
2. Show that X³ + X + 1 | X⁷ + 1 and that K = {0, 1, a, a², a³, a⁴, a⁵, a⁶}.
3. The element a is a zero of X³ + X + 1 (viewed as polynomial in K[X]). Find all zeros.
4. Find the zeros of X³ + X² + 1.