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# Math 521 Lecture 3 Homework 3

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 ex 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 ex 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
x

5. ln x

6. sin x

7. cos x

8. derivation