Download Exercises MAT2200 spring 2013 — Ark 7 Rings and Fields

Exercises MAT2200 spring 2013 — Ark 7
Rings and Fields
This “Ark” concerns the weeks No. �� (Apr ��–��) and No. �� (Apr ��–��).
On Wednesday Apr �� : I’ll do Section 23—Factorisation of Polynomials over a
Field —Section 26—Homomorphisms and Factor Rings—Section 27—Prime and
Maximal Ideals.
On Friday Apr ��: Exercises.
On Wednesday Apr ��: We do Section 29—Introduction to Extension Fields—
Section 31—Algebraic Extensions—Section 32—Geometric Constructions.
On Friday Apr: ��: Exercises.
The exercises on this sheet cover Section 18, Section 19, Section 20, Section 21
and Section 22 in the book. They are ment for the groups on Friday Apr �� and ��
and Thursday Apr �� and �� with the following distribution:
Friday, Apr 19: No.: 4, 5, 7, 8, 11, 12, 16, 17, 19, 21.
Friday, Apr 26: No.: 23, 26, 27, 28, 31, 34, 35, 37.
Thursday, Apr 18: No.: 1, 2, 3, 6, 9, 10, 13, 14, 15, 18, 20.
Thursday, Apr 25: No.: 22, 24, 25, 29, 30, 32, 33, 36.
Key words: Rings, fields, integral domains, units, nilpotents, zero divisors, congruences.
Rings, homomorphisms and isomorphisms
Problem 1. ( Section 18, No.: 1, 2, 3, 5 on page 174 in the book). Compute the
products indicated below:
12 · 16 in Z24 ,
16 · 3 in Z12 ,
11 · ( 4) in Z15 ,
(2, 3) · (3, 5) in Z5 ⇥ Z9
Problem 2. ( Section 18, No.: 23, 24, 25 on page 175 in the book). Describe all
ring homomorphisms
a) from Z to Z,
b) from Z to Z ⇥ Z,
c) from Z ⇥ Z to Z.
Problem 3. Show that the fields R and C are not isomorphic.
Problem 4. (The Gaussian integers). Let Z[i] = { a + bi | a, b 2 Z }. Show that Z[i]
is a subring of the complex numbers.
Rings and Fields
MAT2200 — Vår 2013
p
p
Problem 5. Show that p
Z[ 5] = {pa + b 5 | a, b 2 Z
of the real numbers
p} is a subring
p
R. Show that the map Z[ 5] ! Z[ 5] given by a+b 5 to a b 5 is a ring isomorphism.
Problem 6. Recall that Z[X] denotes the ring of polynomials in the variable X with
integral coefficients. Let
: Z[X] p
! R be the evaluation map sending a polynomial
p
p
P (X) to its value P ( 5) at X = 5. Show that the image of is the subring Z[ 5].
Problem 7. (The binomial theorem). Let R be a commutative ring. Convince yourself
that your favorite proof of the binomial theorem works for R, i.e., prove that if a, b 2 R,
then
n ✓ ◆
X
n n i i
n
(a + b) =
a b.
i
i=0
This is a 100% commutative statement: Indeed, show that if a and b are elements in a
not necessarily commutative ring, then (a + b)2 = a2 + 2ab + b2 if and only if a and b
commute.
Problem 8. (Chinese residue theorem). Assume that n and m are two relatively prime
natural numbers. Define the map : Znm ! Zn ⇥ Zm by ([a]nm ) = ([n]n , [a]m ), where
we let [b]k stand for the residue class of the integer b mod the natural number k. Show
that is a ring isomorphism. Hint: Show that is injective and use that the two
rings have the same number of elements. Take a look at the corresponding statement
for cyclic groups, which we proved some time ago.
Problem 9. Let R✓ M2 (R) be the subset consisting of matrices of the form
✓
◆
x y
y x
where x and y are real numbers.
a) Show that R is a subring of M2 (R)
b) Show that R is a field.
c) Show that R is isomorphic to the field of complex numbers C.
The characteristic of a ring
Problem 10. (Basically Section 19, No.: 6, 8, 10 on page 182 in the book). Determine the characteristic of the following rings
Z3 ⇥ Z3 ,
Z6 ⇥ Z15 ,
Z ⇥ Z,
Q
Problem 11. Show that if p is a prime number, then the binomial coefficients pi
satisfy pi ⌘ 0 mod p. Hence in a commutative ring R of characteristic p, it holds true
that
(a + b)p = ap + bp
for any elements a and b from R.
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Rings and Fields
MAT2200 — Vår 2013
Problem 12. Show that the characteristic of an integral domain is either zero or a
prime number.
Nilpotents, zero divisors and integral domains
Problem 13. What are the nilpotent elements of Z24 ? What are the zero divisors?
Problem 14. Which of the following rings are integral domains?
R = Z9
R = Z11
R = Z13
R = Z15 ?
For each R which is not an integral domain, describe all the zero divisors.
Problem 15. Let R1 and R2 be two rings. Show that the direct product R1 ⇥ R2 is
not an integral domain.
Problem 16. Show that if a and b are elements in the commutative ring R and
an = bm = 0, then (a + b)n+m 1 = 0. Conclude that the sum of two nilpotent elements
is nilpotent. Hint: Use the binomial theorem, problem 7
Problem 17.
a) Show that if R is a finite ring and a 2 R is not a zero divisor, then a is invertible.
Hint: Consider the map R ! R given by x 7! ax.
b) Show by an example that the hypothesis of R being finite is essential.
c) Show that if R is a finite integral domain, then R is a field.
Units and Euler’s -function
Problem 18. ( Section 20, No.: 2, 3 on page 189 in the book). Find the units in the
rings Z11 and Z17 . Show that they are cyclic and exhibit a generator for each of them.
Problem 19. Find the units in the ring Z10 and show that Z⇤10 is cyclic of order 4.
Problem 20. Find the units in the ring Z9 and show that Z⇤9 is cyclic of order 6
generated by the residue class of 2.
Problem 21. Find the units in the ring Z8 and show that Z⇤8 ' Z2 ⇥ Z2 .
Problem 22. Let R1 and R2 be two rings both having a unit element and let R =
R1 ⇥ R2 be their direct product. Show that R⇤ = R1⇤ ⇥ R2⇤ .
Problem 23. Assume that n and m are two natural numbers that are relatively prime.
a) Show that Z⇤nm ' Z⇤n ⇥ Z⇤m . Hint: Use exercise 22 and 8.
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Rings and Fields
MAT2200 — Vår 2013
b) Show that the Euler’s -function satisfies (mn) = (n) (m) whenever n and m
are relatively prime. Hint: Use a).
c) Show that if p is a prime number and r a natural number, then (pr ) = pr 1 (p 1) .
Hint: Count the numbers between 1 and p that are relatively prime to pr .
d) Show that for any natural number n Euler’s -function satisfies
(n)
=
n
Y
(1
p prime and p|n
1
).
p
Problem 24. Show that the group Z[i]⇤ of units in the ring Z[i] of Gaussian integers
(see exercise 4) equals Z[i]⇤ = µ4 = {±1, ±i}.
Squares and square roots
Problem 25. List all the squares in the following rings R and decide in each case if
1 has a square root in R:
R = Z3 ,
R = Z5 ,
R = Z7 .
In which of the rings does X 2 + 2 = 0 have a solution?
Problem 26. Assume that p is an odd prime number.
a) Show that ±1 are the only to elements in Zp with square equal to 1.
that x2 1 = (x 1)(x + 1) in any field.
Hint: Use
b) Regard the group homomorphism : Z⇤p ! Z⇤p sending a to a2 . Show that Ker
{±1}. How many elements are there in the image of ?
=
Problem 27. (Wilsons theorem). Let G be an abelian group written multiplicatively.
Assume that there is just one element
a in G of order 2. Show that the product of all
Q
elements in G equals a; that is x2G x = a.
Apply this to G = Z⇤p , where p is a prime, to show Wilsons theorem
(p
1)! ⌘
1
mod p.
Hint: Problem 26 a) might be useful.
Problem 28. In this exercise p denotes an odd prime. The aim of this exercise is to
give a criterion for when an element a 2 Zp has a square root in Zp , i.e., for when
there is a b 2 Zp with b2 = a.
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Rings and Fields
a) Let
MAT2200 — Vår 2013
: Z⇤p ! Z⇤p be the group homomorphism given by
(a) = a
Show that Im
p 1
2
.
= {±1}. Hint: Use Fermat’s little theorem and exercise 26 a).
b) Show that if a has a square root in Zp , then a 2 Ker .
theorem.
c) Show that a has a square root in Zp if and only if a
26 b).
p 1
2
= 1.
Hint: Fermat’s little
Hint: use problem
d) Show that 1 has a square root in Zp if and only if p ⌘ 1 mod 4, i.e., if and only
if p is of the form p = 4k + 1.
Congruences and equations
Problem 29. Solve the congruences
5x ⌘ 7
mod 13
2x ⌘ 6
mod 4
22x ⌘ 5
mod 15
Problem 30. Show that n and m are two relatively prime natural numbers and
a, b 2 Z, then the simultaneous congruences
x ⌘ a mod m
x ⌘ b mod n
can be solved.
Hint: Use problem 8.
Problem 31. ( Section 19, No.: 1 on page 182 in the book). Find all solutions to
x3 2x2 3x = 0 in Z12 .
Problem 32. ( Section 19, No.: 2 on page 182 in the book). Solve the equation
3x = 2 in the fields Z7 and Z23 .
Problem 33. (Basically Section 19, No.: 3 on page 182 in the book). Find all
solutions of x2 + 2x + 2 = 0 in Z5 in Z6 , and in Z7
Problem 34. ( Section 19, No.: 4 on page 182 in the book). Find all solutions of
x2 + 2x + 4 = 0 in Z6 .
Problem 35. [Equations of second degree]Let R be an integral domain in which 2 is
invertible. Consider the quadratic equation
x2 + bx + c = 0,
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(c)
Rings and Fields
MAT2200 — Vår 2013
where a and b are elements in R. Show by the (usual) procedure of completing the
square that (c) is equivalent to the equation:
(x + b/2)2 = b2 /4
c.
Show that the equation (c) has a solution in R if and only if b2 4c has a square
root in R, and in that case, the solutions are given by the following (usual) formula:
p
b ± b2 4c
x=
.
2
Problem 36. Show that the equation X 2 I = 0, where I denotes the identity
matrix, has infinitely many solutions in the ring M2 (R) of real two by two matrices.
Hint: Check that
✓
◆
0
1
X=
1 0
is a solution and consider the conjugates AXA
1
of X.
Problems from old exams
Problem 37. Eksamen MAT2200, Onsdag 2. juni, 2010 No.: 2.
Versjon: Friday, April 19, 2013 7:58:48 AM
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