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
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Chapter2: Second-order Linear Odes
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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, a∈ F}.
Show that both F[[X]] and F((X)) are integral domains. 
Prove that {F[[X]])× = {a0 + a1X + a2X2 + . . . | ai ∈ F}

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