Rings and fields

Additional exercises

The exercises below have been ordered according to the last section of the book that is needed
in order to be able to do the entire exercise, but often the first few parts are already good
problems for earlier sections of the book.

Section 7.1 (basic definitions)

1. Let n ≥ 2.

(a) Determine all units and all zero divisors of Z/nZ.

(b) Show that Z/nZ is a field if and only if n is a prime number.

2. Let X be a non-empty set.

(a) Prove that the set R of functions from X to the real numbers R is a ring under pointwise
addition and multiplication of functions.
(b) Does this still hold if we replace R with an arbitrary ring A?

3. Let ω = e2πi/3 in C and define the set R = {a + bω with a, b in Z} in C.

(a) Show that R is a subring of C.

(b) Define the norm N on R by N (a + bω) = (a + bω)(a + bω). Show that N takes values
in Z, and that N (αβ) = N (α)(β) for all α and β in R. Does N (α + β) = N (α) + N (β)
also hold for all α and β in R?
(c) Determine the units of R.

4. Let A be a group under addition (so it is assumed to be Abelian), and

R = {f : A → A with f a group homomorphism} .

For f and g in R, we define their sum f +g and their product f ∗g by (f +g)(a) = f (a)+g(a)
and (f ∗ g)(a) = f (g(a)) for a in A.

(a) Show that both f + g and f ∗ g are in R if f and g in R.

(b) Prove that R with the operations + and ∗ is a ring with identity.
(c) Describe the units of R.

5. For α = a + bi + cj + dk in H, we let α = a − bi − cj − dk and N : H∗ → R>0 by N (α) = αα.

Let V = {a + bi + cj + dk in H with a = 0}. Prove that if N (α) = 1 then αvα−1 is in V if v
is in V . Hint: use V = {β in H with β = −β}.
[One can check this describes a rotation on the 3-dimensional R-vector space V and that all
such rotations can be obtained this way. The advantages of using this method over matrices
are that fewer multiplications are involved and that using an approximation α̃ of α will still
result in a rotation. By contrast, an approximation of a rotation matrix may not result in
a rotation. This is disturbing when visualing objects in 3-space, e.g., in computer games
(where computational speed is also important).]

1
 Section 7.2 (polynomial rings, matrix rings, and group rings)
   
1 i 1 j
1. Determine if the matrices and are units and/or zero divisors of M2 (H).
j k i k

2. Let R be a ring (which may or may not be commutative, or not have a 1), X a variable, and
let S = R[X]. We fix an r in R and define the substitution map φ = φr : S → R by mapping
f (X) = a0 + a1 X + a2 X 2 + . . . to a0 + a1 r + a2 r2 + . . . .

(a) Prove that for all f and g in S, we have φ(f + g) = φ(f ) + φ(g).
(b) Show the following: if r is in the centre Z(R) of R, then φ(f · g) = φ(f ) · φ(g) for all f
and g in S.
(c) Show that if R has a 1, then φ(f · g) = φ(f ) · φ(g) for all f and g in S if and only if r is
in Z(R).

3. Let R be a ring, and n ≥ 1. Show that the matrix ring Mn (R) is commutative if and only if

• R is commutative (n = 1);
• every multiplication in R gives 0R (n ≥ 2).

Section 7.3 (ring homomorphisms, and quotient rings)

1. Let R be a ring.

(a) Show that the inverse of a ring isomorphism φ : R → R is itself a ring isomorphism.
(b) Prove that the set with as elements the ring isomorphisms f : R → R is a group under
composition of functions. (This group is denoted Aut(R).)

2. Show that for an element a in the centre of a ring R the map f : R[X] → R given by
f (p(x)) = p(a) is a surjective ring homomorphism.

3. Let φ : R → S be a ring homomorphism, and let X be a variable. Prove that there is a ring
homomorphism

R[X] → S[X]
f (X) 7→ f φ (X)

where, if f (X) = i ai X i , we let f φ (X) = i φ(ai )X i (so we apply φ to the coefficients).

P P

4. Let R be a ring with identity 1R ̸= 0R . Then the map φ : Z → R with φ(m) = m1R is a
ring homomorphism (this follows from 12R = 1R and the rules for calculating with multiples
in an additive group). Its kernel is an ideal (n) of Z for a unique n ≥ 0, which we call the
characteristic of R, denoted char(R).

(a) Prove that if R has no zero divisors (e.g., if it is an integral domain or a field), then
char(R) = 0 or char(R) is a prime number.
(b) Now assume that R is commutative, and char(R) = p with p a prime number. Prove
that Frp : R → R given by mapping a to ap is a ring homomorphism, i.e., a ring
endomorphism of R, the Frobenius endomorphism in characteristic p. What does the
identity Frp (a − b) = Frp (a) − Frp (b) become?

2
 (c) Under the assumption in (b), for every d ≥ 1 the iteration Frdp = Frp ◦ · · · ◦ Frp : R → R
is a ring endomorphism of R. What is the explicit formula for Frdp , and what do the
identities Frdp (a + b) = Frdp (a) + Frdp (b) and Frdp (a − b) = Frdp (a) − Frdp (b) become?

5. Let R = {f : [0, 1] → R with f continuous}.

(a) Show that R, with pointwise addition and multiplication of functions, is a ring.
(b) Prove that for X ⊆ [0, 1] the set IX = {f in R with f (x) = 0 for all x in X} is an ideal
of R.
(c) Is every ideal of R of the form IX as in (b)?

6. Let V = R × R × · · · = {(a1 , a2 , . . . ) with all ai in R}. With coordinatewise multiplication

by scalars, and coordinatewise addition of vectors, this is an R-vector space.

(a) Show that R = {f : V → V with f linear} is a ring if, for f and g in R, we define their
sum f + g by (f + g)(v) = f (v) + g(v), and their product by (f g)(v) = f (g(v)) for every v
in V . (So f g is the composition f ◦ g.) Hint: the proof that R is a ring uses only the
formal properties of V and not its explicit shape.
(b) Determine for the following two subsets of R if it is a left ideal, right ideal, ideal and/or
subring of R.
• I = {f in R such that f (v) has finitely many non-zero coordinates for every v in V }
• J = {f in R with im(f ) finitely generated as R-vector space}

7. Let k be a field, and n ≥ 1 an integer. Show that the only (two sided) ideals of R = Mn (k)
are the ideals {0} and R.

8. (a) Let R = M2 (R). Show that for an R-subspace V of R2 the subset

IV = {A in R such that Av = 0 for all v in V }

is a left ideal of R, and that this gives a bijection between subspaces of R2 and left ideals
of R. Does this result change if we replace R with an arbitrary field F ?
(b) How can you describe all left ideals of the ring Mn (F ) where n ≥ 3 and F is a field?

Section 7.4 (properties of ideals)

1. Let R be a commutative ring with 1, and ideals I = (a1 , . . . , am ) and J = (b1 , . . . , bn ).

Prove that I + J = (a1 , . . . , am , b1 , . . . , bn ).
2. Let R be a commutative ring with 1. Show that (a, b) = (a − qb, b) for all a, b and q in R.
3. Prove that there is a ring isomorphism Z[i]/(2 − 3i) ∼
= Z/13Z. Is (2 − 3i) a maximal ideal
and/or prime ideal of Z[i]?
4. Show that there is a ring isomorphism Z[X]/(n, f (X)) ∼ = Z/nZ[X]/(f¯(X)). Here n is
¯
an integer, X a variable, and f (X) the image of f (X) under the reduction map Z[X] →
Z/nZ[X].
5. Let n be an integer. Prove that the rings Z[i]/(n), Z[X]/(n, X 2 +1) and Z/nZ[X]/(X 2 +1)
are isomorphic.

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 6. Let k be a field, X and Y variables.
∼
(a) Use the first isomorphism theorem in order to show that k[X, Y ]/(Y ) → k[X]. Hint:
why is the map k[X, Y ] → k[X] given by f (X, Y ) 7→ f (X, 0) a ring homomorphism?
∼
(b) Use the second isomorphism theorem in order to show that k[X] → k[X, Y ]/(Y ).
Hint: use S = k[X] and I = (Y ). Why is S + I = k[X, Y ]?
7. (a) Show that in Z we have: (a) ⊆ (b) if and only if b|a.
(b) Now prove that the maximal ideals of Z are the ideals (p) with p a prime number.
Hint: the ideals of Z are (n) with n ≥ 0, and this gives every ideal of Z exactly once.
8. Determine the prime ideals of Z.
9. Let S = Q with the usual addition, but multiplication a · b = 0 for all a and b in S. Prove
that S has no maximal ideals. Hint: for a maximal ideal M , and n ≥ 2, consider the
inclusions M ⊆ n1 M ⊆ S where n1 M = { bn
a
with ab in M }.
10. Let k be a field, and V a k-vector space. As in Exercise 6 here in Section 7.3, the set R
of k-linear maps of V to itself is a ring if we, for f and g in R, define their sum f + g
by (f + g)(v) = f (v) + g(v) for v in V , and their product f g as the composition f ◦ g.
Let I = {f in R such that im(f ) is finitely generated as k-vector space}.
(a) Show that I is an ideal of R.
(b) Show that I is a maximal ideal of R if V has countably infinite dimension as a k-vector
space.

Section 7.5 (rings of fractions)

1. Let D = {2n with n in Z≥0 } = {1, 2, 4, 8, . . . }. Determine all units in the ring D−1 Z.

2. Show that Frac(Z[i]) ≃ Q(i).

√ √
3. Let R = Z[ −3] = {a + b −3 with a and b in Z}, a subring of C. We let

D = {1, 2, 4, 8, . . . } = {2k with k = 0, 1, 2, 3, . . . } ,

√
and let S denote the ring D−1 R. Is 3+ 4 −3 a unit of S?
√ √
4. Let R = Z[ −7] = {a + b −7 with a and b in Z}, and N : R → Z the norm map.

(a) Show that D = {δ in R with N (δ) ≡ 1 modulo 4} satisfies the conditions in Theorem 15
of Section 7.5.
(b) Now let S = D−1 R. Show that S ∗ = { αδ with α in R, δ in D, and N (α) odd}.

Section 7.6 (Chinese remainder theorem for rings)

1. Show that there is a ring isomorphism Z[i]/(13) ∼= Z[i]/(2−3i)×Z[i]/(2+3i). Which element

of Z[i]/(13) is mapped to (3 + (2 − 3i), 6i + (2 + 3i))?

2. Show that there is a ring isomorphism F5 [X]/(X 2 + 1) ∼

= F5 × F5 . Which element is mapped
to (2, 4)?

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 Section 8.1 (Euclidean domains)

1. (a) Determine a generator of the ideal (1599, 943) of Z and write it in the form x1599 + y943
with x and y in Z.
(b) Do the same for the ideal (4371, 2139, 3243) of Z.

2. Determine w and z in Z[i] with w(5 + 4i) + z17 = 2 + 3i.

3. Compute a generator of the ideal (5 + 5i, 8 + 6i) in Z[i], and write it as α(5 + 5i) + β(8 + 6i)
with α and β in Z[i].
√ √ √
4. Show that gcd(2, 1 + −5) = 1 in Z[ −5] but that (2, 1 + −5) ̸= (1). Hint: use the norm
for the first statement, and a ring homomorphism to Z/2Z for the second.
√ √
5. Does gcd(2 − 2 −3, 4) in Z[ −3] exist? If so, what is it, and if not, why not?

6. Compute gcd(9X 4 − 25X 2 + 21X, 3X 2 + 2X − 8) in Q[X]. Are the ideals (9X 4 − 25X 2 + 21X)
and (3X 2 + 2X − 8) of Q[X] comaximal?
√
7. Prove that Z[ −2] is a Euclidean domain. Hint: use the (standard ) norm.

Section 8.2 (principal ideal domains)

1. Let R be a principal ideal domain.

(a) Let x be an irreducible element of R. Show that (x) is a maximal ideal of R. Hint: an
ideal I of R with (x) ⊆ I ⊆ R is of the form (y) for some y in R.
(b) Conclude that:
• every irreducible element of R is a prime element of R;
• every non-zero prime ideal of R is a maximal ideal of R.

Section 8.3 (unique factorisation domains)

1. Factorise 4, 6, 7, 21 − 3i, 13 − 16i, 35 + 7i and 24 + 6i into irreducibles in Z[i].

√
2. Determine all different factorisations into irreducibles of 8 in R = Z[ −3]. For this, we
consider two factorisations 8 = p1 ·· · ··pm and 8 = q1 ·· · ··qn , with all pi and qj irreducible, as
the same if n = m and, perhaps after renumbering the qj , we have q1 = ±p1 , . . . , qm = ±pm .
(Note that R∗ = {±1}.) Hint: use the norm on R.
n o
3. Let x be a variable, and R = fg(x) (x)
in Q(x) with g(0) ̸= 0 . It is given that R is a subring
of Q(x) that is an integral domain.
n o
(a) Show that R∗ = fg(x) (x)
in R with f (0) ̸= 0 .
x
(b) Prove that 1 is an irreducible element of R.
(c) Now show that R is a unique factorisation domain (UFD).

5
 Section 9.2 (polynomial rings over fields I)

1. Let F be a field, r−2 and r−1 two non-zero polynomials in F [X], with deg(r−2 ) ≥ deg(r−1 ).
Put m−2 = n−1 = 1 and n−2 = m−1 = 0. For i ≥ 0, define inductively while ri−1 ̸= 0,

ri = ri−2 − qi ri−1 with deg(ri ) < deg(ri−1 ),

mi = mi−2 − qi mi−1 ,
ni = ni−2 − qi ni−1 .

This is the setup of the extended Euclidean algorithm because the first line determines qi+1
uniquely. Let j be the highest i with ri ̸= 0, so rj is a (possibly non-monic) gcd of r−2 and r−1 ,
and mi r−2 + ni r−1 = ri for i = −2, −1, . . . , j (and, redundantly, mj+1 r−2 + nj+1 r−1 = 0).
Assume that j ≥ 0, i.e., that r−1 does not divide r−2 .

(a) Apply the algorithm with F = F3 , r−2 = X 4 + X and r−1 = X 4 + X 2 + X + 1. How do

the degrees of the qi behave, and how does this influence the degrees of the mi and of
the ni ?
(b) Show that deg(qi ) = deg(ri−2 ) − deg(ri−1 ) for i = 0, 1, . . . , j + 1, and that deg(qi ) > 0
for i = 1, 2, . . . , j + 1.
(c) Prove that deg(mi ) = il=1 deg(ql ) and deg(ni ) = il=0 deg(ql ) for i = 0, 1, . . . , j + 1.
P P
Hint: work out what mi and ni are for i = 0 and 1, before using induction.
(d) Conclude that deg(mi ) = deg(r−1 ) − deg(ri−1 ) and deg(ni ) = deg(r−2 ) − deg(ri−1 ) for
i = 0, 1, . . . , j+1, so that deg(mj ) < deg(r−1 )−deg(rj ) and deg(nj ) < deg(r−2 )−deg(rj ).
(e) Prove that if m̃ and ñ in F [X] satisfy m̃r−2 + ñr−1 = rj , then deg(mj ) < deg(m̃) and
deg(nj ) < deg(ñ) unless (m̃, ñ) = (mj , nj ). Hint: use Exercise 11(b) in Section 9.2.

The above shows that the extended Euclidean algorithm computes the simplest possible
Bézout coefficients if r−1 does not divide r−2 . We now consider the case where r−1 divides r−2 .
Here, it terminates with j = −1, m−1 = 0 and n−1 = 1. Now (d) still applies unless
deg(r−2 ) = deg(r−1 ), i.e., r−2 = cr−1 with c in F ∗ , in which case the simplest Bézout
coefficients are m̃ = d and ñ = 1 − cd for d in F . Finally, we observe that (d) still holds
if r−2 = 0, and that for r−2 ̸= 0 with deg(r−2 ) < deg(r−1 ) the first step swaps the roles of
r−2 and r−1 . So in all cases the algorithm computes simplest possible coefficients, and those
are unique unless r−2 = cr−1 with c in F ∗ .]

Section 9.3 (polynomial rings that are unique factorisation domains)

1. Show that x3 + 3x2 + 2x + 1 is irreducible in Q[x].

2. Prove that x4 + 2x3 + 1 cannot be written as A(x)B(x) with A(x) and B(x) in Q[x] of
degree 2.

Section 9.4 (irreducibility criteria)

1. Formulate and prove the equivalent statement of Proposition 11 for a general unique factori-
sation domain R and its field of fractions F .
√
2. Let R = Z[ −3] and F = Frac(R), and let x be a variable.

6
 √ √
−1− −3
(a) Prove that x2 + x + 1 in R[x] has roots 2 and −1+2 −3 in F .
(b) What are the elements in F of the form rs with r|1 (the constant
term) and s|1 (the
leading term) in R (cf. the statement in Exercise 1)? What does this tell us about R?

3. In this exercise, we find some important variations of Eisenstein’s irreducibility criterion.

(a) Let R be a UFD, with field of fractions F . If f (x) = an xn + · · · + a1 x + a0 for n ≥ 2

is in R[x], and P is a prime ideal of R such that a0 , . . . , an−1 are in P , an is not in P ,
and a0 is not in P 2 , then f (x) is irreducible in F [x]. (Note this is trivial for n = 1.)
(b) Let R be a UFD, with field of fractions F . If f (x) = an xn + · · · + a1 x + a0 for n ≥ 2 is
in R[x], and p is a prime element of R such that a0 , . . . , an−1 are divisible by p, an is
not divisible by p, and a0 is not divisible by p2 , then f (x) is irreducible in F [x].
(c) What is the statement corresponding to Corollary 14 in this general context?

4. Factorise x3 − 7x − 6 and x4 + 4x − 2 in Q[x].

5. Factorise −2x4 + 14x3 + 2x + 22 in Z[x].

6. Factorise xy 2 + x2 − 5xy + y 2 + x − 4y in Q[x, y].

7. Factorise x4 + 3x − 2 in Z[x].

8. Factorise x4 + 3x + 1 in Z[x].

9. Factorise x4 + 3x + 2 in Z[x].

10. Factorise x4 + 3x + 3 in Z[x].

11. Factorise x4 + x3 − 5x2 + 8x − 2 in Z[x].

12. Let k be a field. Factorise y 3 − x2 in k[x, y].

13. Factorise the polynomial into irreducibles in the indicated UFD:

(a) x7 y 2 + xy 8 + y 8 + xy in Q[x, y];

(b) xy 4 + y 4 + x4 y + x2 + x in R[xy];
(c) x2 y 4 + xy 2 + 1 in R[x, y];
(d) x2 y 4 + xy 2 + 1 in C[x, y];
(e) z 5 + xz 3 + yz + y 2 + x in Q[x, y, z].

14. Let F = F2 [y]/(y 3 + y + 1), where y is a variable.

(a) Show that F is a field.

(b) Factorise x3 + x + 1 and x3 + x2 + 1 into monic irreducible factors in F [x]. Hint: if a
in F is a root of such a polynomial, why is then a2 also a root?

15. Let F = F5 [y]/(y 2 + y + 1), where y is a variable.

(a) Show that F is a field.

(b) Factorise x2 + 2 and x2 + x + 2 into monic irreducible factors in F [x].
(c) Factorise x3 + 2 and x3 + x + 1 into monic irreducible factors in F [x].

7
 Section 9.5 (polynomial rings over fields II)

1. Suppose that, in a group, x and y are commuting elements of finite orders m and n respec-
tively. Show that ⟨x, y⟩ contains an element of order lcm(m, n). Hint: first prove that xy
has order lcm(m, n) if gcd(m, n) = 1, next prove the statement by using elements with orders
that are prime powers.