Algebra I – autumn 2023
All your assertions should be carefully justified.
1. (10 points) For each of the following three conditions, decide whether it correctly
defines a subset I of the ring (Z[ 3
√
2], +, ·) and whether this subset is an ideal of this
ring: Let the set I contain the number a + b 3
√
2 + c 3
√
4 + d 3
√
16, with a, b, c, d ∈ Z, if and
only if
(a) [a + b + c]3 = [d]3,
(b) [a + b + d]3 = [c]3,
(c) [a + c + d]3 = [b]3.
2. (10 points) Determine all elements of the transition monoid of the automaton
1
a //
b
882
a,b
// 3
a,b
!!
4
a
aa bee
3. (15 points) Find a direct product of well-known groups that is isomorphic to the
quotient group (G, ·) × (Z, +) /H, where
G =





p 0 0
f 1 0
h g p

 p ∈ {1, −1}, f, g ∈ Z[x], h ∈ Z[i][x]



,
H =







1 0 0
f 1 0
h g 1

 , f(1) + 2k

 ∈ G × Z k, h(2), h(3) ∈ Z, f(1) + g(1) is even



.
4. (10 points) Find the minimal polynomial of the number 4
√
2 + 4
√
4 + 4
√
8 over Q.
5. (15 points) Express the number
1
α5 − 2α4 + α3 + 3α + 1
without using other than rational numbers in denominators, provided that the number
α satisfies the equality α3
(2 − α) = 2α + 2.
6. (10 points) Provide an example of a finite ring (R, +, ·) such that there exist exactly
9
10
|R| elements r ∈ R such that some element s ∈ R \ {0} satisfies the equality r · s = 0.
7. (10 points) Provide an example of a group (G, ·) and two elements g and h of G such
that there exist at least two isomorphisms ϕ: (G, ·) → (G, ·) and each such isomorphism
satisfies both ϕ(g) = g and ϕ(h) = h.
8. (5 points) Define the field of fractions of an integral domain.
9. (5 points) Formulate the theorem that describes irreducible polynomials over the fields
C and R.
10. (10 points) Using only the definitions of groups and subgroups, prove that right cosets
of a subgroup are pairwise disjoint.