Number Theory: An Introduction to Mathematics

266 VI Hensel’sp-adic Numbers Two absolute values,|| 1 and|| 2 ,onafieldFare said to beequivalentwhen, for anya∈F, |a| 1 <1 i ...

2 Equivalence 267 Suppose next that|a|≤1foreverya>1 and so for everya∈Z. Since the absolute value onQis nontrivial, we must h ...

268 VI Hensel’sp-adic Numbers If we puta=b−^1 c,then|a| 1 >1,|a| 2 <1. This proves the assertion form=2. We now assumem> ...

3 Completions 269 It will now be shown that the procedure by which Cantor extended the field of rational numbers to the field of ...

270 VI Hensel’sp-adic Numbers Let(an)be a fundamental sequence which is not inN. Then there existsμ> 0 such that|av|≥μfor inf ...

3 Completions 271 It is easily seen thatF ̄ is uniquely determined, up to an isomorphism which preserves the absolute value. The ...

272 VI Hensel’sp-adic Numbers Proof Lete 1 ,...,enbe a basis for the vector spaceE.Thenanya ∈ Ecan be uniquely represented in th ...

4 Non-Archimedean Valued Fields 273 4 Non-ArchimedeanValuedFields............................... Throughout this section we deno ...

274 VI Hensel’sp-adic Numbers It also follows at once from Lemma 8 that if a sequence(an)of elements ofF converges to a limita= ...

4 Non-Archimedean Valued Fields 275 m/n,wheremandnare relatively prime integers,n>0andpdoes not dividen.The valuation ideal i ...

276 VI Hensel’sp-adic Numbers We now show that, for any nonzeroa∈M, there is a positive integerksuch that |a|=|π|k. In fact we c ...

5 Hensel’s Lemma 277 whereαN,αN+ 1 ,...,αN+n∈Sandan+ 1 ∈R.Since|an+ 1 πN+n+^1 |→0asn→∞, the series ∑ n≥Nαnπ nconverges with suma ...

278 VI Hensel’sp-adic Numbers be a polynomial with coefficients c 0 ,...,cn∈R and let f 1 (x)=ncnxn−^1 +(n− 1 )cn− 1 xn−^2 +···+ ...

5 Hensel’s Lemma 279 |f(ak)|≤θ^2 k 0 σ (^2). Sinceθ 0 <1and|ak+ 1 −ak|=σ−^1 |f(ak)|, this shows that{ak}is a fundamental sequ ...

280 VI Hensel’sp-adic Numbers Conversely, suppose|b− 1 | 2 ≤ 2 −^3. In Proposition 15 takeF = Q 2 and f(x)=x^2 −b. The hypothese ...

5 Hensel’s Lemma 281 Proof Putn=∂(f)andm=∂(φ).Then∂(ψ)=∂(f ̄)−∂(φ)≤n−m.Thereexist polynomialsg 1 ,h 1 ∈R[x], withg 1 monic,∂(g 1 ...

282 VI Hensel’sp-adic Numbers Put gj(x)=xm+ m∑− 1 i= 0 α(ij)xi, hj(x)= n∑−m i= 0 βi(j)xi. By (iii), the sequences(α (j) i )and(β ...

5 Hensel’s Lemma 283 Proposition 19 shows that if a quadratic polynomialat^2 +bt+cis irreducible, then|b|≤max{|a|,|c|}, with str ...

284 VI Hensel’sp-adic Numbers Proposition 21Let F be a complete field with respect to the absolute value||and let the field E be ...

6 Locally Compact Valued Fields 285 is complete, it now follows thatFcontains (a copy of)Rand that the absolute value onFreduces ...