**1 Favourites**

An arsenal of examples in your head is crucial to processing mathematical
concepts. For each

of the following, list your favourite examples. Make sure your list includes
enough examples to

develop an understanding of the concept. If it is not clear that your example is
an example then

prove that it is.

1. sets

2. functions

3. non-functions

4. relations

5. operations

6. monoids

7. groups

8. rings

9. fields

10. division rings

11. integral domains

12. fields of fractions

13. polynomials

14. formal power series

15. equivalence relations

16. rational numbers

17. irrational numbers

18. algebraic numbers

19. transcendental numbers

20. vector spaces

21. algebras

22. derivations

**2 Exercises**

1. Let A be a ring and let a ∈ A. Define 0, −a and a^{-1} and show that 0 ยท a = 0,
0 + 0 = 0,

−(−a) = a, (−1)a = −a, and (a^{-1})^{-1} = a.

2. Show that a finite integral domain is a field.

3. Explain how long division works for polynomials and give some examples.

4. Explain why it is necessary to assume that A is an integral domain when
constructing the

field of fractions of A.

5. Show that the addition operation in the field of fractions is well defined.

6. Show that the multiplication operation in the field of fractions is well
defined.

7. Show that the field of fractions is a field.

8. Let A be an integral domain and let F be the field of fractions of F. Show
that the map

is an injective ring homomorphism.

9. Let A be an integral domain and let F be the field of fractions of F. Show
that if K

is a field with an injective ring homomorphism
then there is a unique
ring

homomorphism such that
.

10. Let F be a field and let a ∈ F. The evaluation homomorphism
F
is defined

from F[x] to F. Discuss thoroughly the issue of extending the evaluation
homomorphism

to F[[x]], F(x), and F((x)).

11. Let S be a set of cardinality n. Show that
is the number of subsets of S of cardinality

k.

12. Show thatis the coefficient of
in (x + y)^{n}.

13. Show that and, if 1 ≤ k ≤ n − 1, then

14. Define e^{x} in 6 different ways and prove that all 6
definitions are equivalent.

15. Define ln x in at least 3 different ways and prove that your definitions are
equivalent.

16. Define sin x in at least 3 ways and prove that your definitions match each
other.

17. Explain why . What does
mean? Where do these expressions

live?

18. Let n ∈ Z>0. Explain why . What does

mean? Where do these expressions live.

19. Let n ∈ Z>0. Define
. Prove that
.

20. Define e^{x} and prove that . Where do these expressions live?

21. Let G = {p(x) ∈ F[[x]] | p(0) = 1}. Show that G is an abelian group under
multiplication.

22. Let g = {p(x) ∈ F[[x]] | p(0) = 0}. Show that g is an abelian group under
addition.

23. Let G = {p(x) ∈ F[[x]] | p(0) = 1} and g = {p(x) ∈ F[[x]] | p(0) = 0}. Show
that

is an isomorphism of groups.

24. Show that .

25. Show that there is a unique derivation
of F[x] such that .

26. Show that if p ∈ F[x] then

27. Show that if p ∈ F[x] then

28. Show that there is a unique extension of
to a derivation of F(x).

29. Show that there is a unique extension of
to a derivation of F[[x]].

30. Show that there is a unique extension of
to a derivation of F((x)).

31. Show that if p ∈ F[[x]] then

32. Show that if p ∈ F[[x]] then

33. Show that if p ∈ F[[x]] then

**3 Vocabulary
**

Define the following terms.

1. commutative ring

2. integral domain

3. field of fractions

4. e

5. ln x

6. sin x

7. cos x

8. derivation