▁non - arch im ede an ▁field ▁, ▁the ▁ring ▁of ▁restricted ▁power ▁series ▁tens ored ▁with ▁, ▁ ▁is ▁called ▁a ▁T ate ▁algebra , ▁named ▁for ▁John ▁T ate . ▁It ▁is ▁equival ently ▁the ▁sub ring ▁of ▁formal ▁power ▁series ▁ ▁which ▁consists ▁of ▁series ▁conver gent ▁on ▁, ▁where ▁ ▁is ▁the ▁valu ation ▁ring ▁in ▁the ▁algebraic ▁closure ▁. ▁ ▁The ▁maximal ▁spectrum ▁of ▁ ▁is ▁then ▁a ▁rig id - anal yt ic ▁space ▁that ▁models ▁an ▁aff ine ▁space ▁in ▁rig id ▁geometry . ▁ ▁Define ▁the ▁G auss ▁norm ▁of ▁ ▁in ▁ ▁by ▁ ▁This ▁makes ▁ ▁a ▁Ban ach ▁algebra ▁over ▁k ; ▁i . e ., ▁a ▁norm ed ▁algebra ▁that ▁is ▁complete ▁as ▁a ▁metric ▁space . ▁With ▁this ▁norm , ▁any ▁ideal ▁ ▁of ▁ ▁is ▁closed ▁and ▁thus , ▁if ▁I ▁is ▁radical , ▁the ▁quot ient ▁ ▁is ▁also ▁a ▁Ban ach ▁algebra ▁called ▁an ▁aff ino id ▁algebra . ▁ ▁Some ▁key ▁results ▁are : ▁( We ier str ass ▁division ) ▁Let ▁ ▁be ▁a ▁- dist ingu ished ▁series ▁of ▁order ▁s ; ▁i . e ., ▁ ▁where ▁, ▁ ▁is ▁a ▁unit ▁element ▁and ▁ ▁for ▁. ▁Then ▁for ▁each ▁, ▁there ▁exist ▁a ▁unique ▁ ▁and ▁a ▁unique ▁polynomial ▁ ▁of ▁degree ▁ ▁such ▁that ▁ ▁( We ier str ass ▁prepar ation ) ▁As ▁above , ▁let ▁ ▁be ▁a ▁- dist ingu ished ▁series ▁of ▁order ▁s . ▁Then ▁there