MAS400: Exercises 1

Throughout this sheet F will denote an arbitrary field unless otherwise stated.

1. For the following pairs of elements f, g ∈ F [x] find the g.c.d. (h say) of f and g and also find elements a, b ∈ F [x] such that af + bg = h. (You should normalise h so that its leading coefficient is 1.)

(a) f = x^{2} − 2x + 1 and g = x^{3} + 3x^{2} − 7x + 3.

(b) f = x^{3} + 1 and g = x^{4} + 5x^{3} + 3x^{2} − 2x − 1.

(c) f = x^{3} − 2 and g = 2x^{2} + 3x − 1.

(d) f = x^{2} + 5x + 4 and g = 3x^{2} − 2x + 1.

[You may assume that F has characteristic 0, say F = ℚ. It is also interesting to see what divergence we get from the general case when we set F = 𝔽_{p} for p a prime.]

2. Show that the ideal of I = ⟨x, y⟩ of F [x, y] cannot be generated by a single element.

3. For n ⩾ 2 show that the ideal of I = ⟨x_{1}, x_{2}, …, x_{n}⟩ of F [x_{1}, x_{2}, …, x_{n}] cannot be generated by a single element. What is the minimum number of elements required to generate I? Justify your answer.

4. Let n, m ∈ ℕ, with n ⩾ 1. How many n-tuples (a_{1}, …, a_{n}) ∈ ℕ^{n} are there such that ∑_{i=1}^{n} a_{i} = m?