MA3052 Exercise Set 4
Unique Factorization Domains (UFD-s)
1. Let R be a domain. Show that any prime in R is irreducible.
2. Let p be a prime integer. Using Cayley’s theorem for the group Z∗p and its cyclic
subgroups < a >, prove that the polynomial X p−1 − 1 ∈ Zp [X] can be decomposed in Zp [X]
into a product of p−1 degree 1 polynomials (uniquely up to order and multiplication by units).
Hence:
a) If p is a prime integer such that p ≡ 1 (mod 4), prove that
(X 2 + 1)|(X p−1 − 1) in Zp [X].
b) If p is a prime integer such that p ≡ 3 (mod 4), prove that
gcd(X 2 + 1, X p−1 − 1) = 1 in Zp [X].
3. a) Show that 2 is not irreducible in Z[i].
b) Let p be a prime integer such that p ≡ 1 (mod 4). Prove that p is not irreducible,
hence not a prime in Z[i].
c) Let p be a prime integer such that p ≡ 3 (mod 4). Prove that p is prime, hence
irreducible in Z[i].
d) What are all the prime elements in Z[i]?
e) What are all the integers which can be written as a sum of two squares?
√
√
√
4. Show that Z[ −5], Z[ 5] and Z[ −6]
√ are not a UFD-s by factoring some elements
in two ways. Give other examples of rings Z[ d] which are not a UFD-s.
5. Prove that Q[X] is a PID and hence a UFD and hence that Z[X] is a UFD, but not
a PID.
6∗ . Let p be a prime number. Prove that the equation√x2 − 2 = 0 has solutions in Zp iff
p ≡ 1 or 7 (mod 8). Hence find all the prime elements in Z[ 2].
7∗ . Let p be a prime number. Prove that the equation x2 + 3 = 0 has solutions in Zp
⇐⇒ p ≡ 1 (mod 6). Hence or otherwise, describe all the prime elements in Z[ξ], where ξ is a
solution of the equation x2 + x + 1 = 0.
The Chinese Remainder Theorem
1. If I1 , I2 , ...., In are ideals in R such that Ij + Ik = R for any j, k distinct, prove that
I1 ∩ I2 ∩ ... ∩ Ik + Ik+1 = R for all k < n. Use this to prove the isomorphism
R
R
R
∼
× ... × .
=
I1 ∩ ... ∩ In
I1
In
2. Given a, b, c, A, B, C ∈ Q such that a 6= b 6= c 6= 0:
a) Find all polynomials P1 (X) such that P1 (a) = A, P1 (b) = 0, P1 (c) = 0.
b) Using part a) or otherwise, find all polynomials P (X) such that P (a) = A, P (b) =
B, P (c) = C.
1
2
3. Find the reminder when X 2016 is divided by X 3 − 3X 2 + 2X.
4. a) Prove that there is an isomorphism F
Z2016 ∼
= Z7 × Z9 × Z32
and find F −1 (1, 0, 0), F −1 (0, 1, 0), F −1 (0, 0, 1), F −1 (1, 2, 3).
b) Is there an isomorphism between Z2016 and Z7 × Z3 × Z3 × Z2 × Z2 × Z2 × Z2 × Z2 ?
How about Z14 × Z9 × Z16 ?
5. Prove the existence of isomorphisms
∼ Q × Q × Q × Q, (ii)
(i) 4 Q[x]2
=
(x −5x +4)
Q[x,y]
(x2 +y 2 −2,x2 −y 2 )
∼
=Q×Q×Q×Q
and construct their inverses.
Q
6∗ . Let n = si=1 pki i be the prime factorization of the number n, with pi 6= pj for i 6= j.
Define the Euler number
ϕ(n) := |U (Zn )| = |{k ∈ {0, 1, ..., n − 1}; gcd(k, n) = 1}|.
Use the Chinese Remainder Theorem to prove that
s
s
Y
Y
ki
ϕ(n) =
ϕ(pi ) =
pki i −1 (pi − 1).
i=1
i=1
7∗ . Alice and Bob are using the RSA decryption algorithm to communicate securely:
Alice releases a public key which consists of the modulus n = 91 and the public (or encryption)
exponent e = 17. Bob wants to transmit a secret number x. He calculates x17 (mod 91), sending
the result 81 to Alice. Help Alice find x by solving the equation x17 ≡ 81(mod 91).
Hint: Recall that by Cayley’s theorem for the multiplicative group U (Zn ) we have xϕ(n) ≡
1(mod n). Combine this with x17 ≡ 81(mod 91) to get x1 .
The security of the RSA encryption is due to the fact that ϕ(n) is calculated based
on the prime factorization of n. Alice knows the prime factorization of n, but for very large
numbers n, it is hard to find its factors, even though n is known.
8∗ . Solve the following RSA decryption problems:
a) x11 ≡ 309( mod 403) b) x173 = 144( mod 209).
In each case, identify the public key, Bob’s message to Alice and Alice’s decryption
procedure.
9∗ . a) Let I = (X 2 + Y 2 + Z 2 − 2, Z 2 − X 2 − Y 2 ) ⊂ R[X, Y, Z]. Draw the solution set
Z(I) in the 3D space and prove that there is a ring isomorphism
R[X, Y ]
R[X, Y, Z] ∼
R[X, Y ]
×
.
=
I
(X 2 + Y 2 − 1) (X 2 + Y 2 − 1)
b) Let J = (Z − X − Y, Z 2 − X 2 − Y 2 ) ⊂ R[X, Y, Z]. Draw the solution set Z(I) in
the 3D space and prove that there is an injective ring homomorphism
R[X, Y, Z]
F :
−→ R[X, Z] × R[Y, Z].
J
Possibly using the geometric interpretation of the rings above, find Im(F ).