A34315
3Num 06 22498 Level H
LH Number Theory
3Num4 06 16214 Level M
LM Number Theory
May/June Examinations 2023-24
1. (a) Determine the factors of 1—4√—2 in Z[ √—2]. [10]
(b) Let α ∈ Z[ √—2] and suppose that N(α ) is irreducible in Z. Prove that α is irreducible in Z[ √ —2]. [7]
(c) Let R ⊆ C be a ring with the properties:
(P1) Every non-zero non-unit element of R is a product of irreducibles.
(P2) Whenever α ∈ R and π and σ are non-associate irreducible factors of α , then πσ | α .
Answer the following.
(i) Suppose that π , σ1 , . . . , σn ∈ R are irreducible and that π | σ1 ··· σn. Prove that π is associate to σj for some j.
(ii) Prove that every irreducible in R is prime.
(iii) State, without proof, what this implies about R. [8]
2. (a) Use the Euclidean Algorithm to determine all the solutions to
32x ≡ 20 mod 108. [10]
(b) State and prove Fermat’s Little Theorem. [7]
(c) Let p be a prime number and let R = { 1, . . . , p − 1 }. For any d ∈ N and a ∈ R let
V(d) = { a ∈ R | a ≡ xd mod p for some x ∈ R } and
Sa(d) = { x ∈ R | xd ≡ a mod p }.
Suppose that d ∈ N and let h = hcf(d , p −1).
(i) Show that S1(h) ⊆ S1(d).
(ii) By considering the equation dx ≡ h mod p − 1, show that there are non-negative integers A and B such that
dA = h+B(p −1).
(iii) Use (ii) to show that S1(h) = S1(d).
(iv) Show that |Sa(d)| = |S1(d)| for all a ∈V(d).
(v) Show that if v ∈V(d), then
v h/p−1 ≡ 1 mod p.
Deduce that |V(d)| ≤ h/p−1.
(vi) Using what you have proved, show that |S1(d)| = h and |V(d)| = h/p−1
. [8]
3. (a) (i) State Gauss’ Law of Quadratic Reciprocity and use it to evaluate the following Leg- endre symbols:
(ii) Let p be a prime number with p ≡ ±2 mod 5. Use Gauss’ Law of Quadratic Reci- procity to prove that
[10]
For the remainder of this question, we work in the ring
Z[√5 ] = { a+b√5 | a, b ∈ Z }.
Recall that Z[√5] possesses a conjugate function defined by
a+b
√5 = a−b
√5
for all a, b ∈ Z.
Define a sequence r1, r2 , . . . by
r1 = 18 and ri+1 = ri(2) —2 for all i ≥ 1.
Finally, let
τ = 2−
√5.
(b) (i) Let p be a prime number with p ≡ ±2 mod 5. Prove that
αp ≡ α mod p
for all α ∈ Z[ √5 ].
(ii) Prove that
ri = τ2i + τ2i
for all i ≥ 1. [7]
(c) Let n ∈ N, n ≥ 3 and set
M = 2n — 1.
Assume that
rn— 1 ≡ 0 mod M.
Assume further that q is a prime number with
q | M and q ≡ ±2 mod 5.
(i) By considering τ2n—1 + τ2n—1, prove that τ2n ≡ —1 mod q and determine the order of τ modulo q.
(ii) Using (b)(i) or otherwise, prove that M is a prime number. [8]
4. (a) Let R be a UFD and π ∈ R an irreducible.
(i) Let α ∈ R\{ 0 }. What is the definition of ordπ (α )?
(ii) Let α , β ∈ R\{ 0 } and suppose that γ is a highest common factor of α and β . State and prove the formula connecting ordπ (γ) , ordπ (α ) and ordπ (β). [7]
(b) Let x and y be integers that satisfy
y3 = x2 +8.
Work in the ring Z[√−2], let π =
√−2 and let γ be a highest common factor of
x+2 √−2 and x−2 √−2.
(i) Prove that γ is associate to πr for some integer r ≥ 0.
(ii) Using (a) or otherwise, prove that r is a multiple of 3 and then that y
3/π
2r
is the cube of an element of Z[√−2].
(iii) Prove that x+2
√−2 is the cube of an element of Z[√−2].
(iv) Determine the possibilities for x and y.
(c) Let p > 25 be a prime number. Show that the number of integer solutions to
yp = x2 +8
is at most p -3. [8]