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Question 30941: List all the polynomials of degree exactly two in Z3[t]. Which of these are reducible, which are irreducible?
Z stands for the integers
Answer by venugopalramana(3286)   (Show Source):
You can put this solution on YOUR website! COEFFICIENTS SHALL BE,0,1,-1 ONLY...SINCE 2 IS SAME AS -1(MOD3),-2=1(MOD3) AND
-3 OR 3=0(MOD3)
HENCE GENERAL SECOND DEGREE POLYNOMIALIS IS
AT^2+BT+C...PUTTING THE ABOVE VALUES WE GET THE FOLLOWING.REDUCIBILITY IS DETERMINED BY CHECKING WHETHER T=0 OR 1 OR -1 YIELDS ZERO VALUE FOT THE QUADRATIC.IF ZERO OCCURS IT IS REDUCIBLE..OTHERWISE NOT...LEGEND..R..REDCIBLE..NR..NOT REDUCIBLE..
1.T^2-T-1..........NR
2.T^2-T...........R...T=0 GIVES ZERO
3.T^2-T+1....NR
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4.T^2-1.....R AT T=1
5.T^2....R AT T=0
6.T^2+1...NR
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7.T^2+T-1...NR
8.T^2+T...R AT T=0
9.T^2+T+1....R AT T=1...SINCE 3 IS 0(MOD3)
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10.-T^2-T-1.....R AT T=1
11.-T^2-T....R AT T=0
12.-T^2-T+1...NR
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13.-T^2-1.....NR
14.-T^2....R AT T=0
15.-T^2+1...R AT T=1
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16.-T^2+T-1...R AT T=-1....SINCE -3=0(MOD3)
17.-T^2+T...R AT T=0
18.-T^2+T+1...NR
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