Yorkshire won this second round Division One fixture in the LV= Insurance County Championship to jump from bottom to top of the in-play table on 38.5 points.
Brook advanced from 79 overnight to 118 off 165 balls as Yorkshire, replying to a first-innings 134, moved from 159 for five to 308 at tea.
New ball seamer Fisher then wreaked havoc in Somerset’s top order to back up his five for 41 on day one with four for 23, reducing them to 18 for five. They were later bowled out for 141 with one ball remaining in the day after the extra half hour was taken.
Brook, 22, was composed but imposing against an off-colour attack on a pitch with pace and bounce, recording his second Championship century of 2021. Thompson’s was his first fifty of the four-day season, and he later claimed three wickets.
Brook’s reputation as one of brightest batting talents in England was further enhanced. He completed an early morning 57-run partnership for the sixth wicket with Harry Duke and then shared 51 for the seventh with Thompson either side of lunch.
Duke, for nine, was the only morning wicket when bowled by Josh Davey - 179-6. Brook reached his century off 122 balls before lunch, his second Championship ton against Somerset, with his first coming at Emerald Headingley on the day England won the one-day World Cup in July 2019.
After Brook had fallen to Tom Abell’s excellent catch at second slip off Ben Green, 230 for seven in the 80th over, Thompson hoisted two leg-side sixes against the second new ball on the way to a 94-ball fifty.
He also shared 55 for the eighth wicket with David Willey, with Marchant de Lange - 4-55 - later mopping up the tail.
Somerset started their second innings with 44 overs left after tea.
Fisher’s four wickets included Pakistan batsman Azhar Ali caught at second slip and George Bartlett lbw and Tom Abell bowled shouldering arms with successive balls as the score fell to 18 for five in the 11th over.
Thompson and George Hill struck further blows before Green, whose 32 was his side’s highest score of the match, and Jack Leach shared 42 for the eighth wicket, advancing from 69-7.
Dom Bess returned the first-innings favour by getting Leach caught at first slip for 26 - 111 for eight - to bring about the extra half hour.
Thompson later bowled de Lange to wrap up the win.