Algebra I – autumn 2023
All your assertions should be carefully justified.
1. (10 points) Consider the subsets I1, I2, I3 of the ring Z[
√
10][x], with Ij consisting
exactly of those polynomials whose constant coefficient belongs to the set Cj and
whose linear coefficient belongs to the set Lj, where
C1 = { 10a + b
√
10 | a, b ∈ Z, [a]9 = [b]9 },
L1 = { 2c + d
√
10 | c, d ∈ Z, [c]3 = [−d]3 },
C2 = { 5a + b
√
10 | a, b ∈ Z, [a]3 = [−b]3 },
L2 = { 2c + d
√
10 | c, d ∈ Z },
C3 = L3 = { 5a + b
√
10 | a, b ∈ Z, [a]4 = [b]4 }.
For each of the subsets I1, I2, I3, decide whether it is an ideal of the ring Z[
√
10][x].
2. (10 points) Determine all elements of the transition monoid of the automaton
1
b,c
HH
a // 2
a
 b,c
// 3
a,c
((
b

4
c
hh
a,b

3. (15 points) Find a direct product of well-known groups that is isomorphic to the
quotient group (G, ·)/H, where
G =





1 a c
0 1 b
0 0 1

 a, b ∈ Z, c ∈ Z[i]



,
H =





1 2d 2d i + e
0 1 4d
0 0 1

 d, e ∈ Z



.
4. (10 points) Find the minimal polynomial of
√
3 + 1 · i −
√
3 + 1 over Q.
5. (15 points) Express the number
1
α3 + α2 − α
without using other than rational numbers in denominators, provided that the
number α satisfies the equality α4
+ 3α3
= 3 · (1 − α2
) · (α + 1).
6. (10 points) Provide an example of a ring (R, +, ·) that is not a subring of any
field, and a subring of (R, +, ·) that is a field.
7. (10 points) Provide an example of a group (G, ·), an isomorphism ϕ: G → G
and a subgroup H ⊆ G such that ϕ(H) ⊆ H and at the same time ϕ(H) = H.
8. (5 points) What does it mean for an integral domain (R, +, ·) to have the unique
factorization property?
9. (5 points) Describe ideals of polynomial rings over fields. Which of them are such
that the corresponding quotient ring is a field/integral domain?
10. (10 points) Using only the definitions of groups and homomorphisms, prove that
every homomorphism of groups preserves identities and inverses.