VALUATIONS

In this problem set (K, v) will denote a discretely valued field of characteristic zero.

1) Suppose that a = (−1 + \sqrt{−3})/2 ∈ K. What is v(a)? [Justify your answer from first principles.]

2) Give a detailed proof of the existence and uniqueness (up to unique isomorphism) of the completion of (K, v).

3) Suppose that p > 2 and A ∈ GL_{n}(ℤ_{p}) satisfies A ≡ I_{n} mod p. If A has finite order show that A = I_{n}. [Hint: Consider the case that A has prime order q make a binomial expansion of (I_{n} + pB)^{q}. Treat the cases q = p and q ≠ p separately.]

4) Which of the following polynomials are irreducible over ℚ_{5}?

X^{2} + 1, X^{2} + 2, X^{2} + 5.

5) Find an integer x with x^{3} + 4x^{2} + 2x − 2 ≡ 0 mod 625.

6) Show that if F and G are pure polynomials of the same slope in K[X] then F G is also pure of that slope. Deduce that, if the only points on NP (F) with integer coordinates are the end points, then F is irreducible. Explain why this implies Eisenstein’s criterion.

7) Which of the following polynomials are irreducible over ℚ_{5}?

X^{3} + 5X + 25, X^{3} + 5X^{2} + 25, X^{4} + 5X^{2} + 25, X^{4} + 15X^{2} + 25.

8) If p is an odd prime determine modulo p^{3} the monic irreducible factors of X^{3} + 2pX^{2} + pX + p^{2}.

9) How many roots does X^{3} + 25X^{2} + X − 9 have in ℚ_{p} for p = 2, 3, 5, 7?

10) Suppose that (K, v) is complete.

(a) Suppose further that L/K is a finite Galois extension and that α, β ∈ L. If

v(β − σα) < v(β − α)

for all σ ∈ G_{L/K} − G_{L/K(α)}, show that α ∈ K(β).

(b) Suppose F ∈ K[X] has degree d. Show that we can find real numbers A and B such that if G ∈ K[X] has degree d and satisfies v_{0}(F − G) > A then any root β of G satisfies v(β) > B.

(c) Suppose F ∈ K[X] is irreducible of degree d. Show that we can find a constant C such that if G ∈ K[X] has degree d and satisfies v_{0}(F − G) > C then G is also irreducible and K[X]/(G) ≅ K[X]/(F). [This is Krasner’s lemma.]

(d) Find the Galois group of the splitting field of X^{4} − 10X^{2} + 27X + 1 over ℚ_{3}. Justify your answer carefully. [Hint: \sqrt{2} + \sqrt{3} is a root of X^{4} − 10X^{2} + 1.]

11) Let L/ℚ denote the splitting field for X^{3} − 2. For p = 2, 3, 7 describe the extensions of v_{p} to L, and for each such extension w identify the decomposition group G_{w} ⊂ G_{L/ℚ}.

12) [This is an extended problem. The result is originally due to Bremner, Lewis and Morton. I have taken this argument from Cassels. You will need to know that a smooth projective plane curve of degree d over a finite field k has a point over k whenever #k > (d − 1)^{2}(d − 2)^{2}. This is a consequence of the so called Riemann hypothesis for curves over a finite field.]

Consider the plane projective curve

3X^{4} + 4Y^{4} − 19Z^{4}.

Show that it has a point over ℝ, ℚ_{2}, ℚ_{3} and ℚ_{19}. Show also that it has a point over 𝔽_{p} for all prime numbers p and deduce that it has a point over ℚ_{p} for all p.

We will now show that it has no solution over the rational numbers ℚ. Explain why, if it did have a rational solution, it would have a solution (x, y, z) with x, y, z integers with no common factor. Suppose (x, y, z) is such a solution in coprime integers. Show that

(19z^{2} − 8y^{2} − 3x^{2})(19z^{2} + 8y^{2} − 3x^{2}) = 57(x^{2} − z^{2})^{2}.

Show moreover that

19z^{2} + 8y^{2} − 3x^{2} > 0

and that

19z^{2} − 8y^{2} − 3x^{2} ≢ 0

modulo 3 or 19. Also show that (19z^{2} − 8y^{2} − 3x^{2}), (19z^{2} + 8y^{2} − 3x^{2}) and (x^{2} − z^{2}) have no common odd prime factor.

Deduce that we can find integers u and v with either

\begin{aligned}19z^{2} − 8y^{2} − 3x^{2} & = & u^{2} \\ 19z^{2} + 8y^{2} − 3x^{2} & = & 57v^{2} \\ z^{2} − x^{2} & = & uv \end{aligned}

or

\begin{aligned}19z^{2} − 8y^{2} − 3x^{2} & = & 2u^{2} \\ 19z^{2} + 8y^{2} − 3x^{2} & = & 114v^{2} \\ z^{2} − x^{2} & = & 2uv. \end{aligned}

Show that the first three equations have no common solution in ℚ_{3} and the second three have no common solution in ℚ_{2}.