A Brief Introduction to Classical and Adelic Algebraic ... - William Stein

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53. Prove that −9 has a cube root in Q10 using the following strategy (this is a
special case of Hensel’s Lemma, which you can read about in an appendix to
Cassel’s article).
(a) Show that there is an element α ∈ Z such that α 3 ≡ 9 (mod 10 3 ).
(b) Suppose n ≥ 3. Use induction to show that if α1 ∈ Z and α 3 ≡ 9
(mod 10 n ), then there exists α2 ∈ Z such that α 3 2 ≡ 9 (mod 10n+1 ).
(Hint: Show that there is an integer b such that (α1 + b · 10 n ) 3 ≡ 9
(mod 10 n+1 ).)
(c) Conclude that 9 has a cube root in Q10.
54. Compute the first 5 digits of the 10-adic expansions of the following rational
numbers:
13
2 ,
1
389 ,
17
,
19
the 4 square roots of 41.
55. Let N > 1 be an integer. Prove that the series
∞
(−1) n+1 n! = 1! − 2! + 3! − 4! + 5! − 6! + · · · .
n=1
converges in QN.
56. Prove that −9 has a cube root in Q10 using the following strategy (this is a
special case of “Hensel’s Lemma”).
(a) Show that there is α ∈ Z such that α 3 ≡ 9 (mod 10 3 ).
(b) Suppose n ≥ 3. Use induction to show that if α1 ∈ Z and α 3 ≡ 9
(mod 10 n ), then there exists α2 ∈ Z such that α 3 2 ≡ 9 (mod 10n+1 ).
(Hint: Show that there is an integer b such that (α1 + b10 n ) 3 ≡ 9
(mod 10 n+1 ).)
(c) Conclude that 9 has a cube root in Q10.
57. Let N > 1 be an integer.
(a) Prove that QN is equipped with a natural ring structure.
(b) If N is prime, prove that QN is a field.
58. (a) Let p and q be distinct primes. Prove that Qpq ∼ = Qp × Qq.
(b) Is Q p 2 isomorphic to either of Qp × Qp or Qp?
59. Prove that every finite extension of Qp “comes from” an extension of Q, in
the following sense. Given an irreducible polynomial f ∈ Qp[x] there exists an
irreducible polynomial g ∈ Q[x] such that the fields Qp[x]/(f) and Qp[x]/(g)
are isomorphic. [Hint: Choose each coefficient of g to be sufficiently close to
the corresponding coefficient of f, then use Hensel’s lemma to show that g
has a root in Qp[x]/(f).]