Algebra I – autumn 2023 – written exam template
All your assertions should be carefully justified.
1. (10 points) Decide whether . . . is a semigroup/monoid/group/ring/integral do-
main/field.
(for instance, Decide whether (Z, ∗), where ∗ is the operation defined by the rule
a ∗ b = a + b − ab for all a, b ∈ Z, is a semigroup and whether it is a group.)
or
Decide whether . . . is a subsemigroup/submonoid/subgroup/normal subgroup/
subring/ideal of . . . .
2. (10 points) Determine all elements of the transition monoid of the automaton
. . . .
(the automaton can be, for instance,
1
a
ÖÖ b GG 2
a
ÖÖ b GG 3
a,b
33
4
a
—— bee )
3. (15 points) Find a direct product of well-known groups that is isomorphic to the
quotient group (G, ·)/H.
for instance,
(G, ·) =







1 0 0
0 a 0
b c 1

 | a ∈ Q \ {0}, b ∈ C, c ∈ R



, ·


H =





1 0 0
0 a 0
bi c 1

 | a ∈ {−1, 1}, b, c ∈ R



4. (10 points) Find the minimal polynomial of the number . . . over Q.
(the number can be, for instance, 1 + 3
√
2 − 1 · i,
√
3 +
3
√
3 + 3, 3
√
9 − 3
√
3 + 3)
5. (15 points) Express the number 1
...
without using other than rational numbers in
denominators.
(the number can be, for instance, 1
α2−α+1
, where α satisfies α3
+ 2α2
+ 2α = −2)
6. – 7. (2 × 10 points) Provide an example of a semigroup/group/ring/homomorphism
with given properties.
(for instance, a group that contains elements of every possible order or an infinite
group and its subgroup of index 10)
8. (5 points) Define . . . .
9. (5 points) Formulate the theorem . . . .
10. (10 points) Prove . . . .
The answer to each of the questions 8. – 10. was presented during the lectures.