Let F be a field. Two variations of the ring of polynomial over F are the...

 

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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: and F[[X]] = {a0 + a₁X + a²X² + ... . | ai Є F} -k+1 7-k+2 F((X)) = {a_kX¯k + a−k+1X¯¯ +a-k+2X- + . |k Є Z, ai Є F}. ... Show that both F[[X]] and F((X)) are integral domains. (Please, do not submit your work on this part!) (a) Prove that (F[[X]])* = {ao + a1X + a2x² + . Є F[[X]] | ao 0}. (b) Find the inverse of 1+2X in F[[X]]. (c) Prove that F((X)) is a field. ... 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: and F[[X]] = {a0 + a₁X + a²X² + ... . | ai Є F} -k+1 7-k+2 F((X)) = {a_kX¯k + a−k+1X¯¯ +a-k+2X- + . |k Є Z, ai Є F}. ... Show that both F[[X]] and F((X)) are integral domains. (Please, do not submit your work on this part!) (a) Prove that (F[[X]])* = {ao + a1X + a2x² + . Є F[[X]] | ao 0}. (b) Find the inverse of 1+2X in F[[X]]. (c) Prove that F((X)) is a field. ...