Ideals of Z[x] with xZ[x]⊆J⊆Z[x]

Assignment 1
Student Name
The University of Sydney
School of Mathematics and Statistics

1. a) Provide an example and proof for each of the following:
i) An irreducible element in the ring Q[x].
Example: x2+ 1.
Proof:
Suppose that x2+ 1 is reducible in Q[X].
Then the equation x2+ 1=0 has a root in Q. That is, there is a b in Q such that
b2=−1. This implies that b= √−1 is in Q. Clearly this is not true and thus
x2+ 1 is not reducible in Q[X].
ii) A maximal ideal in Z[x].
Example: (3 , x).
Proof:
We need to show that (3 , x) is a maximal ideal. Let us suppose that (3 , x) is
not maximal and thus there is another generator polynomial, q ∉(3 , x ).
Let Q be some polynomial. We can write p¿ k + x∗Q where k is an integer not
divisible by 3. Note that subtracting a multiple of one generator from another
generator does not change the ideal. Thus (3 , x , P)=(3 , x , n). Since n is not a
multiple of 3 , gcd (3 , n)=1 , so 1 ∈(3 , x , P).
Therefore(3 , x , P) is all of Z [x ]→ (3 , x) is maximal.
iii) A unit u ≠ 1 in the ring Z[x]/( x2 + 3x + 1)Z[x].
Since x2 + 3x + 1 irreducible in Z then x2 + 3x + 1 is a unit in Z[x]/(x2 + 3x +
1)Z[x].
iv) Example: 1− √−3.

Proof:
We show that 1− √−3 is irreducible in Z[√−3].
Assume 1− √−3 is reducible, then there must exist a , b ∈ Z [ √ −3] so that N¿
b) Write 45+ 420 i as a product of irreducible Gaussian integers, showing all working.
Solution
In Z [i], N (a+ bi)=a2 +b2 .
N (45 420 i)=452 + 4202=178425=3∗3∗5∗5∗13∗61
Thus 45+ 420 i has factors a+ bi, c+di , s +ti∧m+ ¿ such that
N (a+bi) =32, N (c+ di)=5 , N ( s +ti )=13∧N ( m+¿ )=61.
If these factors exist, then they are clearly irreducible over Z.
We have a+ bi=32=9. 9 cannot be expressed as a sum of two elements of Z and thus
a+ bi=3.
We have c +di=5 which can be expressed as 12 +22 .
We have s+ti=13 which can be expressed as 32 + 42 .
We have m+¿=61 which can be expressed as 52 +62.
Thus 45+ 420 i=(3)(3)(1+ 2i)(1−2 i)(2+i)(2−i)(3+2 i)(3−2i)(5+6 i)(5−6 i)
c) Let R, S, T be rings, and suppose that α : R → S and β : R → T are ring
homomorphisms.
Show that the map γ :R → S ×T with γ (x) = (α(x), β(x)) is a ring homomorphism.
Solution