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Problem Set #7 Due: 24 October 2008 1. Let R := C ([0, 1]) be the ring of continuous real-valued functions on the closed interval [0, 1] ⊂ R. For X ⊆ [0, 1], define I( X ) := { f ∈ R : f ( x ) = 0 for all x ∈ X }. (a) Prove that I( X ) is an ideal of R. (b) If p ∈ [0, 1] and I p = I({ p}), then prove that I p is a maximal ideal of R and R ∼ I p = R. (c) For J ⊂ R, set V( J ) := { x ∈ [0, 1] : f ( x ) = 0 for all f ∈ J }. Prove that V( J ) is a closed subset of [0, 1]. Hint: A function is continuous if and only if the inverse image of any closed set is closed. (d) If I is a proper ideal of R, then show that V( I ) 6= ∅. Hint: A topological space is compact if every open covering has a finite subcover. (e) Prove that any maximal ideal P of R equals I p for some p ∈ [0, 1]. 2. Solve the following system of simultaneous congruences in Q[ x ]: f ( x ) ≡ −3 (mod x + 1) , f ( x ) ≡ 12x (mod x2 − 2) , f ( x ) ≡ −4x (mod x3 ) . 3. For a field K, let K (( x )) be ring of formal Laurent series with coefficients in K: n ∞ o n K (( x )) := ∑ an x : an ∈ K, m ∈ Z is arbitrary j=m where the ring operations are defined as in the ring of formal power series K [[ x ]]. Prove that K (( x )) is isomorphic to the fields of fractions for the domain K [[ x ]]. 4. Let K be a field. Show that there exists a smallest subfield F of K and that F is canonically isomorphic either to Q or F p . MATH 893: page 1 of 1