Download Exercises MAT2200 spring 2013 — Ark 9 Field extensions and

Exercises MAT2200 spring 2013 — Ark 9
Field extensions and Unique factorisation
This is the last Ark!
The plans are as follows. If we don’t have time to do all, we’ll stop where we stop!
Wednesday May �: We do the end of Section 27—Prime fields, Ideal Structure of
F[X]—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 the trial exam on Wednesday May 15, it will be posted on
the web as usual. It will not be corrected, but I’ll do it on the blackboard on Friday
May ��.
The exercises on this sheet cover Section 29, Section 31, Section 32, Section 45,
Section 46 and Section 47 in the book. They are ment for the groups on Wednesday
May �� and Thursday May �� and �� with the following distribution:
Wednesday, May 22: No.: 2, 3, 4, 5, 7, 9, 12, 16, 19, 20.
Thursday, May 23: No.: 1, 8, 10, 11, 13, 14, 15, 17, 18, 19
Key words: Field extensions and Unique factorisation. Old exams.
Field extensions
Oppgave 1. (Basically Section 29, No.: 1 and 2 on page 272 in the book). For each
of the complex numbers ↵ below find a polynomial f (x) 2 Q[x] such that f (↵) = 0.
q
q
p
p
p
p
p
p
↵=1+ 2
↵= 1+ 2
↵= 2+ 3
↵=
2+ 3
Oppgave 2. ( Section 29, No.: 6, 7 and 8 on page 272 in the book). For each of
the complex numbers ↵ below find an irreducible polynomial f (x) 2 Q[x] having ↵ as
a root.
q
q
p
p
p
↵= 3
6
↵ = 1/3 + 7
↵= 2+i
Oppgave 3. Let F be a field of characteristic p > 0 and let Zp ✓ F be the prime field.
Let f (x) = xp x a for an element a 2 F .
Ark9: Field extensions and Unique factorisation MAT2200 — Vår 2013
a) Show that if ↵ is a root of f , then also ↵ + i for i 2 Zp is a root.
Oppgave 4. Let F be a finite field with pn elements and let Zp ✓ F be the prime field.
a) Show that if x 2 F satisfies xp = x, then x 2 Zp Hint: Use Fermat’s little theorem
to find the roots of T p T .
b) Show that xp
n 1
2 Zp for all x 2 F
Hint: Take a look at exercise 3 b) on Ark8.
c) Assume that p = 2. Show that x2 x a is irreducible over F for any non-zero a 2 F .
n 1
n 1
n 1
Hint: If ↵ is a root, show that ↵2 (↵2 + 1) = a2 .
Oppgave 5. Show that the polynomial f (x) = x2 2 2 Z5 [x] is irreducible and
conclude that F = Z5 [x]/(f (x)) is a field in which f has a root. We denote a root by
p
2.
p
a) Show that F = { a + b 2 | a, b 2 Z5 } and that every element in Z5 has a square
root in F .
p
p
p
b) Show that (a + b 2)5 = a b 2 and that (a + b 2)6 = a2 2b2 .
p
c) Explain why F ⇤ is a cyclic group of order 24. Show that an element a + bp2 is a
generator
for F ⇤ if and only if ab 6= 0 and a2 b2 6= 1. Which of the elements 2 + 2 and
p
3 + 2 generate F ⇤ ?
Oppgave 6.
a) Show that f (x) = x3
p is irreducible where p is a prime number.
p
b) Show that the homomorphism Q[x] ! R sending a polynomial g(x) to g( 3 p) induces
p
p
p
an injection Q[x]/(f (x)) into R whose image is Q( 3 p) = { a+b 2 p+c 3 p2 | a, b, c 2 Q }.
c) Let ! be a primitive third root of unity. Show that the homomorphism Q[x] ! C
p
sending a polynomial g(x) to g(! 3 p) induces an injection Q[x]/(f (x)) into C whose
p
p
p
image is Q(! 3 p) = { a + b! 2 p + c! 2 3 p2 | a, b, c 2 Q }.
p
p
d) Show that the two fields Q(! 3 p) and Q( 3 p) are isomorphic fields, but are different
subfields of C.
Oppgave 7. Find a basis over Q for each of the fields below
p
p p
p p
p
p
3
Q( 3) Q( 2, 2) Q( 2, 3) Q( 2 + 3)
p
p
Oppgave 8. Find a basis over Q( 2 + 3) for each of the fields below
p
p
p p
Q( 2)
Q( 6)
Q( 2, 3)
Oppgave 9. ( Section 31, No.: 28
page 292 in the book).pAssume
thatpa and
p
pb
p on p
are two rational numbers such that a + b 6= 0. Show that Q( a, b) = Q( a + b)
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Ark9: Field extensions and Unique factorisation MAT2200 — Vår 2013
Oppgave 10. ( Section 31, No.: 30 on page 292 in the book). Let F ✓ E be a field
extension. Assume that ↵ 2 E is algebraic over F of odd degree. Show that ↵2 is
algebraic of odd degree and that F (↵) = F (↵2 ).
Oppgave 11. Show that the algebraic closure of Q in C is not a finite extension of Q.
Oppgave 12. (Basically Section 31, No.: 35 on page 293 in the book). Show that no
finite field F is algebraically closed. Hint: If the characteristic is odd show there is
an irreducible quadratic polynomial of the form x2 a with a 2 F . If the characteristic
is two, try x2 x a.
Unique factorisation
Oppgave 13. ( Section 45, No.: 9 on page 399 in the book). If possible, give four
different associates of 2x 7 in each of the rings Z[x], Q[x] and Z11 [x].
Oppgave 14. Factor the polynomial 4x2
4x + 8 in the rings Z[x], Q[x] and Z11 [x].
Oppgave 15. Let ⌫(n) be the number of digits (to the base ten)of the integer n. Show
that ⌫ is a Euclidean norm on Z. Is it essential that the base is ten?
p
p
Oppgave 16. For ↵ = x + y
2 2 Q(
2) let N (↵) = |↵|2 = x2 + 2y 2
a) Show that N (↵ ) = N (↵)N ( ).
p
p
b) Show that for any ↵ 2 Q(
2) there is a 2 Z[
2] such that N (↵
)<1
p
c) Show that Z[
2] is a Euclidean domain.
Old exams
Oppgave 17. Exam June 2011 No. 1
Oppgave 18. Exam June 2011 No. 2
Oppgave 19. Exam June 2011 No. 3
Oppgave 20. Exam June 2011 No. 4
Versjon: Monday, January 14, 2013 12:07:03 PM
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