Zachary Scherr
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Math 503 HW 5
Due Friday, Feb 26
Reading
1. Read Chapter 9 of Dummit and Foote
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Problems
1. DF 9.1: 13, 14
2. DF 9.2: 2 (use 7.4.14), 3, 5
3. DF 9.3: 1, 2
4. DF 9.4: 8, 11, 12, 13
5. Prove that the polynomial f (x) = x2 + x + 1 ∈ F5 [x] is irreducible. Find an element in F5 [x]/(x2 + x + 1)
which generates the group of units.
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Challenge Problems
Challenge Problems tend to be harder than the rest of the problems (and sometimes more interesting). You
do not need to turn these in, but you should get something out of thinking about these.
1. Prove that every prime ideal of Z[x] is of one of the following forms
1. (0)
2. (f (x)) where f (x) ∈ Z[x] is an irreducible non-constant polynomial
3. (p) where p ∈ Z is prime
4. (p, g(x)) where p ∈ Z is prime, g(x) ∈ Z[x] is non-constant, and g(x) ∈ (Z/pZ)[x] is irreducible.
Which of the above ideals are maximal?
2. Let R be the set of all functions from the positive integers to the complex numbers:
R = {f : Z+ → C}.
Consider the operations of + and ∗ on R where
(f + g)(n) = f (n) + g(n)
n
X
(f ∗ g)(n) =
f (d)g
d
d|n
(a) Prove that R is a commutative ring under + and ∗ with identity element defined by
(
1, n = 1
(n) =
0, n > 1
(b) Prove that the units in R are precisely those functions with f (1) 6= 0.
(c) A multiplicative function f : Z+ → C is a function satisfying f (ab) = f (a)f (b) whenever gcd(a, b) =
1. Prove that if f, g ∈ R are both multiplicative then so is f ∗ g.
Zachary Scherr
Math 503 HW 5
Due Feb 26
(d) Consider the function 1 ∈ R defined by 1(n) = 1 for every n ≥ 1 and let µ ∈ R be the inverse of 1.
Then g = f ∗ 1 if and only if g ∗ µ = f . Use this to prove that if ϕ(n) is the Euler phi function,
n
X
.
ϕ(n) =
d·µ
d
d|n
(The function µ can be used to prove that there are irreducible polynomials over Fp of degree n for
any n ≥ 1. You can look at exercises 9.5.5 and 9.5.6 to see how already this is useful. This formula
also gives an alternative proof that ϕ is multiplicative, which we’ve previously seen via the Chinese
Remainder Theorem.)
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