By Baker.

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Qs be primes which satisfy the congruences pi ≡ 1 (1 4 i r), qj ≡ 3 4 (1 j s). Show that p1 p2 · · · pr q1 q2 · · · qs ≡(−1)s . 4 Use this result to show that for any natural number n, 4n + 3 is divisible by at least one prime p with p ≡ 3. 4 1-11. (a) Find two roots of the polynomial f (x) = x4 + 22 modulo 23. Hence find three factors of f (x) modulo 23 and explain why you would not expect there to be any other monic linear factors. (b) Find two roots of the polynomial g(x) = x4 + 4x2 + 43x3 + 43x + 3 modulo 47.

Then h is a bijection. d) Plot each pair (a, b) as the point in the xy-plane with coordinates (a, b); such points are all those with natural number coordinates. Starting at (0, 0) we can now trace out a path passing through all of these points and we can arrange to do this without ever recrossing such a point. 4. POWER SETS AND THEIR CARDINALITY .. ··· (0, 3) O (0, 2) o 57 ··· ··· ··· / (1, 3) ··· GG GG GG GG # ··· ··· ··· ··· (1, 2) (2, 2) cGG GG GG GG GG GG GG GG # / (1, 1) ··· (2, 1) (3, 1) cGG cGG DD GG DD GG GG GG DD GG GG GG DD GG GG GG D!

So (−1)cσ +cτ = (−1)cτ σ . A permutation σ is called even if sgn σ = 1, otherwise it is odd. The set of all even permutations in Sn is denoted by An . Notice that ι ∈ An and in fact the following result is true. 8. The set An forms a group under composition. Proof. 7, if σ, τ ∈ An , then sgn(τ σ) = sgn(τ ) sgn(σ) = 1. Note also that ι ∈ An . 32 2. GROUPS AND GROUP ACTIONS The arrow diagram for σ −1 is obtained from that for σ by interchanging the rows and reversing all the arrows, so sgn σ −1 = sgn σ.