Download Exercises MAT2200 spring 2013 — Ark 8 Polynomials, Factor

Exercises MAT2200 spring 2013 — Ark 8
Polynomials, Factor rings and Ideals
There will be no time to start on Galois theory, so we stop at section 45. I’ll give a
detailed curriculum later on.
The plans for May are then as follows:
Wednesday May �: We do Section 29—Introduction to Extension Fields—Section
31—Algebraic Extensions—Section 32—Geometric Constructions.
On Wednesday May ��: We do Section 45—Unique Factorization Domains—Section
46—Euclidean Domains—Section 47—Gaussian Integers and Multiplicative Norms.
On Wednesday May ��: We do exercises.
On Wednesday May ��: Repetisjon.
On Friday May ��: Repetisjon.
Trial exam: I’ll give you a trial exam. It will not be corrected, but I’ll do it on the
blackboard on Friday May ��. On the other Fridays and Thursday we will do exercises
as usual.
The exercises on this sheet cover Section 23, Section 26, and Section 27 in the
book. They are ment for the groups on Friday May � and �� and Thursday May �
and �� with the following distribution:
Friday, May 3: No.: 4, 8, 9, 13, 14, 15, 16, 27, 28.
Friday, May 10: No.: 17, 18, 24, 26, 29, 30.
Thursday, May 2: No.:1, 2, 3, 5, 6, 7, 10.
Thursday, May 16: No.: 11, 12, 19, 20, 21, 22, 23, 25.
Key words: Finite fields, groups of units, polynomials, factor rings, ideals, prime ideals,
maximal ideals.
Finite fields and groups of units
Problem 1. Let K be a finite integral domain. Show that K is a field.
Problem 2. Assume that K is a finite field and let A✓ K be the subring generated
by 1, i.e., A = { m · 1 | m 2 Z }.
a) Show that A is a field and that A ' Zp for some prime p.
b) Show that the additive order of any non-zero element in K equals p. Conclude that
for some natural number n there is an isomorphism K ' Znp as additive groups.
Hint: m · a = (m · 1) · a. Use the classification of finite abelian groups.
c) Conclude that if q denotes the number of elements in the finite field K, then q = pn
for some prime p and some natural number n.
Ark8: Polynomials, Factor rings and Ideals
MAT2200 — Vår 2013
Problem 3. Let K be a finite field with q = pn elements.
a) Show that K ⇤ is a cyclic group of order q 1.
b) (General case of Fermat’s little theorem). Show that aq = a for all a 2 K.
Problem 4. Recall that Euler’s -function is given by (N ) being the number of
integers k with 1  k < N which are relatively prime to N , or equivalently, the
number of elements in the group Z⇤N of units in ZN .
a) Let p be a prime. Show that the number of generators of the group of units Z⇤p equals
(p 1).
b) How many generators does Z⇤2011 have? Hint: In problem 23 d) on Ark7 you found
a formula for (N ).
c) Compute the number of generators of Z⇤p for p a prime and p  23.
Problem 5. ( Section 23, No.: 5, 6, 7 and 8 on page 218 in the book). Find all the
generators of Z⇤p in the following cases
Z⇤5
Z⇤7
Z⇤17
Z⇤23 .
Factoring polynomials over finite fields
Problem 6. ( Section 23, No.: 2 and 4 on page 218 in the book). In each of the
cases below find the quotient and the remainder when dividing f (X) by g(X):
f (X) =X 6 + 3X 5 + 4X 2 3X + 2 and
f (X) =X 4 + 5X 3 3X 2 and
g(X) = 3X 2 + 2X 3 in Z7 [X]
g(X) = 5X 2 X + 2 in Z11 [X].
Problem 7. (Basically Section 23, No.: 9 on page 218 in the book). Show that
X 4 + 4 splits as a product of linear factors in Z5 [X].
Problem 8. Show that a quadratic polynomial X 2 + bX + c is irreducible in Zp [X] if
and only if b2 4c is a square in Zp . Hint: Take a look at problem 35 on Ark7.
Problem 9. Let f (X) = X 3 + 2X 2 + 2X + 1.
a) ( Section 23, No.: 10 on page 218 in the book). The polynomial f splits as a
product of linear factors in Z7 [X]. Find the factorisation. Hint: 3 is a square in Z7 .
b) Let p be an odd prime. Show that g(X) = X 2 + X + 1 splits a product of linear
factors in Zp [X] is and only if 3 is a square in Zp .
c) Does f split as a product of linear factors over Z11 ?
Problem 10. ( Section 23, No.: 12 and 13 on page 218 in the book). In each of
the following two cases, decide whether the polynomial f (X) is irreducible in Z5 [X].
In case it is not, exhibit a factorization.
f (X) = X 3 + 2X + 1
f (X) = 2X 3 + X 2 + 2X + 2
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Ark8: Polynomials, Factor rings and Ideals
MAT2200 — Vår 2013
Miscellaneous
Problem 11. Let N be a natural number. For any polynomial f 2 Z[X], let f be
the polynomial
in ZN [X] obtained by
P
Preducing the coefficients of f modulo N , i.e., if
f (X) = ri=0 ai X i , we let f (X) = ri=0 [ai ]N X i where [a]N denotes the residue class
of a mod N .
Show that f + g = f + g and f g = f g. In other words, the map Z[X] ! ZN [X]
sending f to f is a ring homomorphism.
Problem 12. Let K✓
which is not a
pC be a fieldpand let d 2 K denote an element
p
square i K. We let K( d) = { a + b d | a, b 2 K }. Show that K( d) is a field.
p p
p
p
p
Problem 13. Show that the set
Q(
2,
3)
=
{
a
+
b
2
+
c
3
+
d
p
p p
p 6 | a, b, c, d 2 Q }
is a field. Hint: Let K = Q( 2). Show that Q( 2, 3) = K( 3) and use problem
12.
Factoring polynomials over Z and Q
Problem 14. Let A 2 Z be an integer with A 6= ±2. Show that X 4 + AX 3 + 1 is
irreducible in Z[X].
Problem 15. Let f (X) = X 4 + AX 2 + 1 with A 2 Z.
a) Assume that f (X) is not irreducible. Show that there is a factorization
f (X) = (X 2
aX + d)(X 2 + aX + d)
where d = ±1 and a 2 Z.
b) Show that if f is not irreducible, then A = 2
Conclude that if A > 1, then f is irreducible.
a2 or A =
2
a2 for some a 2 Z.
c) Show that f (X) = X 4
22X + 1 is irreducible in Z[X].
d) Show that f (X) = X 4
Z[X].
23X + 1 is not irreducible and find a factorisation of f in
Problem 16. Assume that f (X) is a monic polynomial in Z[X], i.e., the leading
coefficient og f equals one. Let p be a prime, and let f¯(X) 2 Zp [X] be the polynomial
obtained by reducing the coefficients of f modulo p.
a) Let f (X) = X 4 + 22X 3 + 12 and p = 11. Show that f¯ = X 4 + 1.
b) Assume that f¯ is irreducible in Zp [X]. Show that f is irreducible in Z[X].
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Ark8: Polynomials, Factor rings and Ideals
MAT2200 — Vår 2013
Problem 17. Let p be an odd prime and let a and b be two nonzero elements in Zp .
Show that either a, b or ab has a square root in Zp . Hint: Use Problem 28 c on Ark
p 1
7 where it is shown that an element a 2 Z⇤p is a square if and only if a 2 = 1.
Problem 18.
a) Show that the polynomial f (X) = X 4
10X 2 + 1 is irreducible in Z[X].
b) Show that we have the equality
p
Hint: Square both sides.
3±
p
q
p
2 = 5 ± 2 6.
c) Show that the following three equalities hold:
p
p
X 4 10X 2 + 1 = (X 2 2 2X 1)(X 2 + 2 2X 1)
p
p
= (X 2 2 3X + 1)(X 2 + 2 3X + 1)
p
p
= (X 2 (5 + 2 6))(X 2 (5 2 6))
d) Why does this notpcontradict
the unique factorization of polynomials withpcoefficip
p
ents over the field Q( 2, 3)? See problem 13 for a definition of the field Q( 2, 3).
e) Show that f¯ is reducible in Zp [X] for any prime p.
above and problem 17 with a = 2 and b = 3.
Hint: Use the factorizations
Ideals and factor rings
Problem 19. ( Section 26, No.: 2 on page 243 in the book). Describe all ideals in
Z12 . In each case describe the factor ring Z12 /I.
Problem 20. If K is a field and : K ! R is a homomorphism into a ring R with
unity, show that f is injective. (We assume that (1) = 1).
Problem 21. Let
a) Assume that
: R ! R0 be homomorphism between the two rings R and R0 .
is surjective. Prove that (I) is an ideal in R0 .
b) Show by an example that the hypothesis of
being surjective in a) is essential.
c) Let J✓ R0 be an ideal in R0 . Show that the inverse image
J } is an ideal in R.
1
(J) = { x 2 R | (x) 2
Problem 22. Assume that p is a prime and let b and c be elements in Zp . Show that
Zp [X]/(X 2 + bX + c) is a field if and only if b2 4c is not a square in Zp
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Ark8: Polynomials, Factor rings and Ideals
MAT2200 — Vår 2013
Problem 23. ( Section 27, No.: 6 and 7 on page 252 in the book). For which
elements c in Z3 are the following factor rings fields?
Z3 [X]/(X 3 + X 2 + c)
Z3 [X]/(X 3 + cX + 1)
Problem 24. Which of the following two rings are fields?
Q[X]/(X 2
5X + 5)
Q[X]/(X 2
5X + 6)
Problem 25. Assume that R is a commutative ring with 1 and that I✓ R is an ideal
such that R/I is finite. Show that I is prime if and only if I is maximal.
Problem 26. Show that A = Z7 [X]/(X 2 + 1) is a field. Let i denote the residue class
of X. Show that i2 = 1 and that any element in A can be written in a unique way
as x = a + bi with a and b from Z7 . This motivates that we rebaptize A as Z7 (i). How
many elements does Z7 (i) have? Is i a square in Z7 (i)?
Problems from old exams
Problem 27. Eksamen MAT2200, 6. juni, 2008 No. 1
Problem 28. Eksamen MAT2200, 6. juni, 2008 No. 2
Problem 29. Eksamen MAT2200, 6. juni, 2012 No. 2
Problem 30. Eksamen MAT2200, 2. juni, 2010 No. 3a an 3b.
Versjon: Friday, April 19, 2013 7:40:27 AM
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