Math 254A. Homework 12

Due Wednesday, 15 November

1(nc). Find at least two roots of the polynomial x^{3} − 13x + 4 in ℚ_{2} . It will suffice to give inductive definitions of sequences converging to the given roots, and to show that the two roots are different. [Hint: use the method of the handout on Hensel’s lemma.]

2. Describe the subgroup (ℚ_{7}^{∗})^{2} of ℚ_{7}^{∗} explicitly in terms of p-adic series

a_{m}p^{m} + a_{m+1}p^{m+1} + ….

(Just describe the set, not the binary operation or inverse operation.)

3(nc). Show that the book’s Hensel lemma (II (4.6)) implies the special case of the version of

Hensel’s lemma given in class on Friday, November 3.

(This special case says the following. Let K = (K, | · |) and A be as in the handout on Hensel’s lemma, let f ∈ A[x] , and let α_{0} ∈ A. Assume that |f(α_{0})| < 1 and |f′(α_{0})| = 1 . Then there exists α ∈ K such that f(α) = 0 and |α − α_{0}| < 1 .)

4. Show that there is a unique quadratic extension of ℚ_{2} whose valuation ring has residue field 𝔽_{4} .

5. Let ξ be algebraic of degree n over ℚ_{p}. Show that there is an integer N such that ξ does not satisfy any congruence

a_{n−1}ξ^{n−1} + a_{n−2}ξ^{n−2} + ⋯ + a_{0} ≡ 0 (mod p^{N}) ,

in which the a_{i} are rational integers, not all divisible by p. [Hint: Compactness.]