Let F be a field. Two variations of the ring of polynomial over F are the ring of formal power series F[[X]] and the ring of formal Laurent series F((X)) defined as follows: F[[X]] = {a0 + a1X + a2X2 + . . . | ai ∈ F} and F((X)) = {a−kX−k + a−k+1X−k+1 + a−k+2X−k+2 + . . . |k ∈ Z, ai ∈ F}. Show that both F[[X]] and F((X)) are integral domains. Prove that {F[[X]])× = {a0 + a1X + a2X2 + . . . | ai ∈ F}

Let F be a field. Two variations of the ring of polynomial over F are the ring of formal power series F[[X]] and the ring of formal Laurent series F((X)) defined as follows: F[[X]] = {a0 + a1X + a2X2 + . . . | ai ∈ F} and F((X)) = {a−kX−k + a−k+1X−k+1 + a−k+2X−k+2 + . . . |k ∈ Z, ai ∈ F}. Show that both F[[X]] and F((X)) are integral domains. Prove that {F[[X]])× = {a0 + a1X + a2X2 + . . . | ai ∈ F}

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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