Course 311: Academic Year 2001-02

1. Let x be an integer, and let p be a prime number. Suppose that x^{3} ≡ 1 (mod p). Prove that either x ≡ 1 (mod p) or else x^{2} + x ≡ −1 (mod p).

2. Let x be a rational number. Suppose that x^{n} is an integer for some positive integer n. Explain why x must itself be an integer.

3. Find a function f: ℤ^{3} → ℤ with the property that f(x, y, z) ≡ x (mod 3), f(x, y, z) ≡ y (mod 5) and f(x, y, z) ≡ z (mod 7) for all integers x, y, z.

4. Is 273 a quadratic residue or quadratic non-residue of 137?

5. Let p be a prime number. Prove that there exist integers x and y coprime to p satisfying x^{2} + y^{2} ≡ 0 (mod p) if and only if p ≡ 1 (mod 4).

6. Let p be a odd prime number, and let g be a primitive root of p.

(a) Let h is an integer satisfying h ≡ g (mod p). Explain why the order of the congruence class of h modulo p^{2} is either p − 1 or p(p − 1). Hence or otherwise prove that h is a primitive root of p^{2} if and only if h^{p−1} ≢ 1 (mod p^{2}).

(b) Use the result of (a) to prove that there exists a primitive root of p^{2}. (This primitive root will be of the form g + kp for some integer k.)

(c) Let x be an integer, and let m be a positive integer. Use the binomial theorem to prove that if x ≡ 1 (mod p^{m}) and x ≢ 1 (mod p^{m+1}) then x^{p} ≡ 1 (mod p^{m+1}) and x ≢ 1 (mod p^{m+2})

(d) Use the results of previous parts of this question to show that any primitive root of p^{2} is a primitive root of p^{m} for all m ≥ 2. What does this tell you about the group of congruence classes modulo p^{m} of integers coprime to p?

(e) Do the above results hold when p = 2 (i.e., when the prime number p is no longer required to be odd)?

7. Let G be a group. An automorphism of G is an isomorphism sending G onto itself. Show that the set Aut(G) of automorphisms of G is a group with respect to the operation of composition of automorphisms.

8. Let G be a group. The centre Z(G) of G is defined by

Z(G) = {z ∈ G : gz = zg for all g ∈ G}.

Prove that the centre Z(G) of a group G is a normal subgroup of G. [In particular, you should show that Z(G) is a subgroup of G.]

9. Let H be a subgroup of a group G. The normalizer N(H) of H in G is defined by N(H) = {g ∈ G : gHg^{−1} = H}. Verify that N(H) is a subgroup of G and H is a normal subgroup of N(H).

10. (a) Show that the elements of the alternating group A_{5} fall into five conjugacy classes, and calculate the number of elements in each conjugacy class. Verify that the sum of the numbers obtained equals the order of A_{5}.

(b) Any normal subgroup of A_{5} is a union of conjugacy classes. Show how information on the sizes of the conjugacy classes of A_{5} can be combined with Lagrange’s Theorem to show that the group A_{5} is simple.

11. (a) Show that the alternating group A_{5} has 10 subgroups of order 3. Show also that any two of these subgroups are conjugate.

(b) Show that the alternating group A_{5} has 5 subgroups of order 4. Show also that any two of these subgroups are conjugate.

(c) Show that the alternating group A_{5} has 6 subgroups of order 5. Show also that any two of these subgroups are conjugate.

12. Use Eisenstein’s criterion to verify that the following polynomials are irreducible over ℚ:—

(i) t^{2} − 2; (ii) t^{3} + 9t + 3; (iii) t^{5} + 26t + 52.

13. Let p be a prime number. Use the fact that the binomial coefficient

\binom{p}{k} is divisible by p for all integers k satisfying 0 < k < p to show

that if tf(t) = (t + 1)^{p} − 1 then the polynomial f is irreducible over ℚ.

The cyclotomic polynomial Φ_{p}(t) is defined by Φ_{p}(t) = 1+t+t^{2} +⋯+ t^{p−1} for each prime number p. Show that tΦ_{p}(t + 1) = (t + 1)^{p} − 1, and hence show that the cyclotomic polynomial Φ_{p} is irreducible over ℚ for all prime numbers p.

14. The Fundamental Theorem of Algebra ensures that every non-constant polynomial with complex coefficients factors as a product of polynomials of degree one. Use this result to show that a non-constant polynomial with real coefficients is irreducible over the field ℝ of real numbers if and only if it is either a polynomial of the form at + b with a ≠ 0 or a quadratic polynomial of the form at^{2} + bt + c with a ≠ 0 and b^{2} < 4ac.

15. Let f_{1}, f_{2}, …, f_{k} be non-constant polynomials with coefficients in a field K, and let g = f_{1}f_{2} ⋯ f_{k} + 1. Show that g is not divisible by f_{1}, f_{2}, …, f_{k}. Use this result to show that there are infinitely many irreducible polynomials with coefficients in a field K.

16. A complex number z is said to be algebraic if there f(z) = 0 for some non-zero polynomial f with rational coefficients. Show that z ∈ ℂ is algebraic if and only if ℚ(z): ℚ is a finite extension Then use the Tower Law to prove that the set of all algebraic numbers is a subfield of ℂ.

17. Let K, L and M be fields satisfying K ⊂ L ⊂ M. Suppose that the field extensions M: L and L: K are algebraic (but not necessarily finite). Prove that the extension M: K is algebraic.

18. Let L be a splitting field for a polynomial of degree n with coefficients in K. Prove that [L: K] ≤ n!.

19. (a) Show that ℚ(\sqrt{2}, \sqrt{3}) = ℚ(\sqrt{2} + \sqrt{3}) and [ℚ(\sqrt{2}, \sqrt{3}), ℚ] = 4. What is the degree of the minimum polynomial of \sqrt{2} + \sqrt{3} over ℚ?

(b) Show that \sqrt{2}+\sqrt{3} is a root of the polynomial t^{4}−10t^{2}+1, and thus show that this polynomial is an irreducible polynomial whose splitting field over ℚ is ℚ(\sqrt{2}, \sqrt{3}).

(c) Find all ℚ-automorphisms of ℚ(\sqrt{2}, \sqrt{3}), and show that they constitute a group of order 4 isomorphic to a direct product of two cyclic groups of order 2.

20. Let K be a field of characteristic p, where p is prime.

(a) Show that f ∈ K[t] satisfies Df = 0 if and only if f(t) = g(t^{p}) for some g ∈ K[t].

(b) Let h(t) = a_{0} + a_{1}t + a_{2}t^{2} + ⋯ + a_{n}t^{n}, where a_{0}, a_{1}, …, a_{n} ∈ K. Show that (h(t))^{p} = g(t^{p}), where g(t) = a_{0}^{p} + a_{1}^{p}t + a_{2}^{p}t^{2} + ⋯ + a_{n}^{p}t^{n}.

(c) Now suppose that Frobenius monomorphism of K is an automorphism of K. Show that f ∈ K[t] satisfies Df = 0 if and only if f(t) = (h(t))^{p} for some h ∈ K[t]. Hence show that Df ≠ 0 for any irreducible polynomial f in K[t].

(d) Use these results to show that every algebraic extension L: K of a finite field K is separable.

21. A field K is said to be algebraically closed if every non-constant polynomial with coefficients in K splits over K. Use the fact that the number of irreducible polynomials with coefficients in a given field K is infinite to prove that any algebraically closed field must be infinite.

22. For each positive integer n, let ω_{n} be the primitive nth root of unity in ℂ given by ω_{n} = exp(2πi/n), where i = \sqrt{−1}.

(a) Show that the field extensions ℚ(ω_{n}): ℚ and ℚ(ω_{n}, i): ℚ are normal field extensions for all positive integers n.

(b) Show that the minimum polynomial of ω_{p} over ℚ is the cyclotomic polynomial Φ_{p}(t) given by Φ_{p}(t) = 1 + t + t^{2} + ⋯ + t^{p−1}. Hence show that [ℚ(ω_{p}): ℚ] = p − 1 if p is prime.

(c) Let p be prime and let α_{k} = ω_{p^{2}}ω_{p}^{k} = exp(2πi(1 + kp)/p^{2}) for all integers k. Note that α_{0} = ω_{p^{2}} and α_{k} = α_{l} if and only if k ≡ l mod p. Show that if θ is an automorphism of ℚ(ω_{p^{2}}) which fixes ℚ(ω_{p}) then there exists some integer m such that θ(α_{k}) = α_{k+m} for all integers k. Hence show that α_{0}, α_{1}, …, α_{p−1} all belong to the orbit of ω_{p^{2}} under the action of the Galois group Γ(ℚ(ω_{p^{2}}): ℚ(ω_{p})). Use this result to show that [ℚ(ω_{p^{2}}): ℚ(ω_{p})] = p and [ℚ(ω_{p^{2}}): ℚ] = p(p − 1).

23. Show that the field ℚ(ξ, ω) is a splitting field for the polynomial t^{5} − 2 over ℚ, where ω = ω_{5} = exp(2πi/5) and ξ = ^{5} \sqrt{2}. Show that [ℚ(ξ, ω): ℚ] = 20 and the Galois Γ(ℚ(ξ, ω): ℚ) consists of the automorphisms θ_{r,s} for r = 1, 2, 3, 4 and s = 0, 1, 2, 3, 4, where θ_{r,s}(ω) = ω^{r} and θ_{r,s}(ξ) = ω^{s}ξ.

24. Let f be a monic polynomial of degree n with coefficients in a field K.

Then

f(t) = (t − α_{1})(t − α_{2}) ⋯ (t − α_{n}),

where α_{1}, α_{2}, …, α_{n} are the roots of f in some splitting field L for f over K. The discriminant of the polynomial f is the quantity δ^{2}, where δ is the product ∏_{1≤i<j≤n}(α_{j} − α_{i}) of the quantities α_{j} − α_{i} taken over

all pairs of integers i and j satisfying 1 ≤ i < j ≤ n.

Show that the quantity δ changes sign whenever α_{i} is interchanged with α_{i+1} for some i between 1 and n − 1. Hence show that θ(δ) = δ for all automorphisms θ in the Galois group Γ(L: K) that induce even permutations of the roots of f, and θ(δ) = −δ for all automorphisms θ in Γ(L: K) that induce odd permutations of the roots. Then apply the Galois correspondence to show that the discriminant δ^{2} of the polynomial f belongs to the field K containing the coefficients of f, and the field K(δ) is the fixed field of the subgroup of Γ(L: K) consisting of those automorphisms in Γ(L: K) that induce even permutations of the roots of f. Hence show that δ ∈ K if and only if all automorphisms in the Galois group Γ(L: K) induce even permutations of the roots of f.

25. (a) Show that the discriminant of the quadratic polynomial t^{2} + bt + c is b^{2} − 4c.

(b) Show that the discriminant of the cubic polynomial t^{3} − pt − q is 4p^{2} − 27q^{2}.

26. Let f(t) = t^{3} − pt − q be a cubic polynomial with complex coefficients p and q, and let the complex numbers α, β and γ be the roots of f.

(a) Give formulae for the coefficients p and q of f in terms of the roots α, β and γ of f, and verify that α + β + γ = 0 and

α^{3} + β^{3} + γ^{3} = 3αβγ = 3q

(b) Let λ = α + ωβ + ω^{2}γ and μ = α + ω^{2}β + ωγ, where ω is the complex cube root of unity given by ω = \frac{1}{2}(−1 + \sqrt{3}i). Verify that 1 + ω + ω^{2} = 0, and use this result to show that

α = \frac{1}{3}(λ + μ),  β = \frac{1}{3}(ω^{2}λ + ωμ),  γ = \frac{1}{3}(ωλ + ω^{2}μ).

(c) Let K be the subfield ℚ(p, q) of ℂ generated by the coefficients of the polynomial f, and let M be a splitting field for the polynomial f over K(ω). Show that the extension M: K is normal, and is thus a Galois extension. Show that any automorphism in the Galois group Γ(M: K) permutes the roots α, β and γ of f and either fixes ω or else sends ω to ω^{2}.

(d) Let θ ∈ Γ(M: K) be a K-automorphism of M. Suppose that

θ(α) = β, θ(β) = γ, θ(γ) = α.

Show that if θ(ω) = ω then θ(λ) = ω^{2}λ and θ(μ) = ωμ. Show also that if θ(ω) = ω^{2} then θ(λ) = ωμ and θ(μ) = ω^{2}λ. Hence show that λμ and λ^{3} + μ^{3} are fixed by any automorphism in Γ(M: K) that cyclically permutes α, β, γ. Show also that the quantities λμ and λ^{3} + μ^{3} are also fixed by any automorphism in Γ(M: K) that interchanges two of the roots of f whilst leaving the third root fixed. Hence prove that λμ and λ^{3} + μ^{3} belong to the field K generated by the coefficients of f and can therefore be expressed as rational functions of p and q.

(e) Show by direct calculation that λμ = 3p and λ^{3} + μ^{3} = 27q. Hence show that λ^{3} and μ^{3} are roots of the quadratic polynomial t^{2} − 27qt + 27p^{3}. Use this result to verify that the roots of the cubic polynomial t^{3} − pt − q are of the form

3 \sqrt{\frac{q}{2} + \sqrt{\frac{q^{2}}{4} − \frac{p^{3}}{27}}} + ^{3} \sqrt{\frac{q}{2} − \sqrt{\frac{q^{2}}{4} − \frac{p^{3}}{27}}}

where the two cube roots must be chosen so as to ensure that their product is equal to \frac{1}{3}p.