Can some one help with me this ? I am not sure on how to do this.

please write the logic and solution clearly. hope nice writing.

here is the hw

MAT 150C: MODERN ALGEBRA

Homework 4

Instructions. Please write the answer to each problem, including the computational ones, in connected

sentences and explain your work. Just the answer (correct or not) is not enough. Write your name in every

page and upload to Gradescope with the correct orientation. Make sure to indicate to Gradescope which

pages correspond to each problem. Finally, if you used another sources or discussed the problem

with classmates, be sure to acknowledge it in your homework.

1. (a) Show that if F is a field of positive characteristic p > 0, then (a + b)p = ap + bp for every a, b ∈ F.

(b) Let p be a prime number and r > 0 an integer. Let Φp (x) = xp−1 + xp−2 + · · · + x + 1 be the

cyclotomic polynomial. Show that the polynomial

r−1

Φpr (x) := Φp (xp

r−1

)=

(xp )p − 1

xpr−1 − 1

is irreducible in Z[x]. (Hint: This is very similar to the proof that Φp (x) is irreducible)

2. At the end of class on√Friday 04/17 I said a huge lie and this is your opportunity to correct me. Indeed,

I mentioned that Z[ −5] is not a UFD because 5 ≡ 1 mod

√ 4. But in fact, there is nothing

√ special

about 5. Here we will see that, for every integer n > 2, Z[ −n] is not a UFD. Note that Z[ −n] ⊆ C,

so it is indeed a domain. Take n > 2.

√

√

(a) Define a function

N

:

Z[

−n]

→

Z

by

N

(α

+

β

−n) = α2 + nβ 2 . Show that N (ab) = N (a)N (b)

√

for a, b ∈ Z[ −n].

√

(b) Show, using

part

(a),

that

a

∈

Z[

−n] is a unit if and only if N (a) = 1. Use this to find all the

√

units in Z[ −n].

√

(c) Use parts (a) and (b) to show that 2 ∈ Z[ −n] is irreducible. (Hint: If 2 = ab is a decomposition,

then N (2) = N (a)N (b))

√

√

(d) Assume n is odd. Show that N (1 + −n) is divisible by 2, and use this to find a, b ∈ Z[ −n]

such √

that 2 divides ab but 2 does

√ not divide a nor b. This tells us that 2 is not a prime element

in Z[ −n]. Conclude that Z[ −n] is not a UFD.

(e) Now √

assume n is even. Use a similar procedure to part (d) to show that 2 is not a prime element

in Z[ −n].

√

3. We have seen in class (04/20) that Z[ −1] is a Euclidean

domain, therefore a PID and therefore a

√

UFD. Adapt the proof we saw in class to show that Z[ −2] is a Euclidean domain as well.

1