Tirohia te wāhanga i raro nei, kātahi ka whakautu i te pātai i muri mai i te reo kotahi me te wāhanga:  Passage: Ko te nuinga o te wā e whai ana nga kaikaikarau ki te whakarōpūtanga oti (ranei te rārangi) o tētahi ariā pāngarau. I roto i te horopaki o ngā rōpū mutunga, ka arahina tēnei whāinga ki ngā pāngarau uaua. E ai ki te kaupapa a Lagrange, ko nga roopu mutunga o te raupapa p, he tau matua, he roopu huringa (Abelian) Zp. Ka taea hoki te whakaatu ko nga roopu o te raupapa p2 he Abelian, he korero kaore i te whanui ki te raupapa p3, e whakaatu ana te roopu kore-Abelian D4 o te raupapa 8 = 23 i runga ake nei. Ka taea te whakamahi i ngā pūnaha taurangi rorohiko hei whakarārangi i ngā rōpū iti, engari kāore he whakarōpū o ngā rōpū mutunga katoa. Ko te taahiraa takawaenga ko te whakarōpūtanga o ngā rōpū māmā mutunga. Ka kīia he rōpū kāore i te mea noa iho mēnā ko ōna rōpū iti noa iho ko te rōpū noa iho me te rōpū anō. Ko te kaupapa a Jordan <unk> Hölder e whakaatu ana i nga roopu ngawari mutunga hei waahanga hanga mo nga roopu mutunga katoa. Ko te whakarārangi i ngā rōpū māmā katoa he whakatutukitanga nui i roto i te ariā rōpū o nāianei. I angitu a Richard Borcherds, toa o te Fields Medal 1998, ki te whakamatau i nga whakaaro whakamataku o te marama, he hononga ohorere me te hohonu i waenga i te roopu tino nui - te "roopu whakamataku" - me etahi mahi modular, he waahanga o te tātari matatini tawhito, me te ariā aho, he ariā e whakaarohia ana hei whakakotahi i te whakaahuatanga o nga ahuatanga tinana maha.  Pātai: He aha tētahi o ngā hiahia o te Mathematicians?
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