Math 662 Homework 2 Spring 2014 Drew Armstrong Problem 0. (Drawing Pictures) The equation y 2 = x3 − x defines a “curve” in the complex “plane” C2 . What does it look like? Unfortunately we can only see real things, so we substitute x = a + ib and y = c + id with a, b, c, d ∈ R. Equating real and imaginary parts then gives us two simultaneous equations: (1) a3 − a − 3ab = c2 − d2 , (2) b3 + b − 3a2 b = −2cd. These equations define a real 2-dimensional surface in real 4-dimensional space R4 = C2 . Unfortunately we can only see 3-dimensional space so we will interpret the b coordiante as “time”. Sketch the curve in real (a, c, d)-space at time b = 0. [Hint: It will look 1-dimensional to you.] Can you imagine what it looks like at other times b? Problem 1. (Local Rings) Let R be a ring. We say R is local if it contains a unique (nontrivial) maximal ideal. (a) Prove that R is local if and only if its set of non-units is an ideal. (b) Given a prime ideal P ≤ R, prove that the localization na o : a, b ∈ R, b 6∈ P RP := b is a local ring. [Hint: The maximal ideal is called P RP .] (c) Consider a prime ideal P ≤ R. By part (b) we can define the residue field RP /P RP . Prove that we have an isomorphism of fields: Frac(R/P ) ≈ RP /P RP . [Hint: The most obvious map R/P → RP /P RP must factor through Frac(R/P ).] Problem 2. (Formal Power Series) Let K be a field and consider the ring of formal power series: K[[x]] := a0 + a1 x + a2 x2 + a3 x3 + · · · : ai ∈ K for all i ∈ N . The “degree” of a power series does not necessarily exist. However, for all nonzero f (x) = P k k k a x we can define the “order” ord(f ) := the minimum k such that ak 6= 0. (a) Prove that K[[x]] is a domain. (b) Prove that K[[x]] is a Euclidean domain with norm function ord : K[[x]] − {0} → N. (You can define ord(0) = −∞ if you want.) [Hint: Given f, g ∈ K[[x]] we have f |g if and only if ord(f ) ≤ ord(g), so the remainder is always zero.] (c) Prove that the units of K[[x]] are just the power series with nonzero constant term. (d) Conclude that K[[x]] is a local ring. (e) Prove that Frac (K[[x]]) is isomorphic to the ring of formal Laurent series: K((x)) := a−n x−n + a−n+1 x−n+1 + a−n+2 x−n+2 + · · · : ai ∈ K for all i ≥ −n . Problem 3. (Partial Fraction Expansion) To what extent can we “un-add” fractions? Let R be a PID. Consider a, b ∈ R with b = pe11 pe22 · · · pekk where p1 , . . . , pk are distinct primes and e1 , . . . , ek ≥ 1. (a) Prove that there exist a1 , . . . , ak ∈ R such that a a2 ak a1 = e1 + e2 + · · · + e1 . b p1 p2 pk [Hint: First prove it when b = pq with p, q coprime. Use B´ezout.] Now assume that R is a Euclidean domain with norm function N : R − {0} → N. (b) Prove that there exist c, rij ∈ R such that k e i XX rij a , =c+ b pji i=1 j=1 where for all i, j we have either rij = 0 or N (rij ) < N (pi ). [Hint: If p is prime, prove q that we can write pae as pe−1 + pre where either r = 0 or N (r) < N (p). Then use (a).] Now suppose that the norm function satisfies N (a) ≤ N (ab) and N (a−b) ≤ max{N (a), N (b)} for all a, b ∈ R − {0}. (c) Prove that the partial fraction expansion from part (b) is unique. [Hint: Suppose we have two expansions c+ ei k X X rij i=1 j=1 pji k = e 0 i XX rij a = c0 + . b pji i=1 j=1 Then we get a partial fraction expansion of zero: ei k X 0 ) X (rij − rij a−a 0 0 = = (c − c ) + . b b pji i=1 j=1 For all i, j define ˆbij := b/pji , 0 so that b(c − c) = ei k X X 0 ˆ (rij − rij )bij . i=1 j=1 0 and let j be maximal Suppose for contradiction that there exist i, j such that rij 6= rij 0 ) and with this property. Use the last equation above to show that pi divides (rij − rij hence 0 0 N (pi ) ≤ N (rij − rij ) ≤ max{N (rij ), N (rij )} < N (pi ). Contradiction.] (d) If K is a field and R = K[x] then the norm function N (f ) = deg(f ) satisfies the hypotheses of part (c) so the expansion is unique. Compute the unique expansion of x5 + x + 1 ∈ R(x). (x + 1)2 (x2 + 1) (e) If R = Z then the norm function N (a) = |a| does not satisfy |a − b| ≤ max{|a|, |b|}. However, if we require remainders r, r0 to be nonnegative then it is true that |r − r0 | ≤ max{|r|, |r0 |} and the proof of uniqueness in (c) still goes through. Compute the 77 unique expansion of 12 ∈ Q with nonnegative parameters rij ≥ 0.

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