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Question 31362: I= {h(t)*([4]t^4 + [2]t^3 + [6]t^2 + [4]t + [5]) + k(t)*([3]t^3 + [5]t^2 + [6]t):h(t), k(t) belong to Z7[t]} which is a subset of Z7[t].
a) how do i show that I is an ideal of Z7[t]
b) find a polynomial d(t) belongs Z7[t] such that I=(d(t)), i.e. I is the principle ideal generated by d(t).
Answer by venugopalramana(3286)   (Show Source):
You can put this solution on YOUR website! I= {h(t)*([4]t^4 + [2]t^3 + [6]t^2 + [4]t + [5]) + k(t)*([3]t^3 + [5]t^2 + [6]t):h(t), k(t) belong to Z7[t]} which is a subset of Z7[t].
LET A AND B BE 2 ELEMENTS OF I
a) how do i show that I is an ideal of Z7[t]
LET A AND B BE 2 ELEMENTS OF I
WE HAVE TGO SHOW THAT A+B IS ALSO AN ELEMENT OF I FOR I TO BE AN IDEAL OF Z7(T)
LET A ={h1(t)*([4]t^4 + [2]t^3 + [6]t^2 + [4]t + [5]) + k1(t)*([3]t^3 + [5]t^2 + [6]t)}
LET B={h2(t)*([4]t^4 + [2]t^3 + [6]t^2 + [4]t + [5]) + k2(t)*([3]t^3 + [5]t^2 + [6]t)}
A+B={[h1(t)+H2(T)]*([4]t^4 + [2]t^3 + [6]t^2 + [4]t + [5]) + [k1(t)+K2(T)]*([3]t^3 + [5]t^2 + [6]t)}
SINCE H(T) AND K(T) BELONG TO Z7(T),{H1(T)+H2(T)}=H'(T) SAY AND {K1(T)+K2(T)}=K'(T) SAY ALSO BELONG TO Z7(T)
SO A+B={h'(t)*([4]t^4 + [2]t^3 + [6]t^2 + [4]t + [5]) + k'(t)*([3]t^3 + [5]t^2 + [6]t)}.....BELONGS TO I
HENCE I IS AN IDEAL OF Z7(T)
b) find a polynomial d(t) belongs Z7[t] such that I=(d(t)), i.e. I is the principle ideal generated by d(t).
WE HAVE IT FROM THE ABOVE THAT
P=[4]t^4 + [2]t^3 + [6]t^2 + [4]t + [5])...AND...Q=[3]t^3 + [5]t^2 + [6]t) ARE THE 2 MULTIPLERS OF H(T) AND K(T) TO GET I .HENCE THE GENERATOR IS GIVEN BY GCD OF THESE 2 MULTIPLIERS P AND Q
SINCE IF THEIR GCD IS G THEN IT FOLLOWS THAT,G=XP+YQ....FOR SOME X AND Y ELEMENTS OF Z7(T) AND HENCE FOR ANY ELEMENT A IN I
A={h(t)*([4]t^4 + [2]t^3 + [6]t^2 + [4]t + [5]) + k(t)*([3]t^3 + [5]t^2 + [6]t)}
WE CAN FIND ONE X AND Y IN Z7T,SUCH THAT A CAN BE EXPRESSED IN THE DESIRED FORM.
SO GCD OF P AND Q IS
Q=3T^3+5T^2+6T=T(3T^2+5T+6)
SINCE 3T^2+5T+6=14=0MOD(7) FOR T=1....SO ...T-1 IS A FACTOR IN Z7
P=T(T-1)(3T+1)
P=4T^4+2T^3+6T^2+4T+5...=21=0MOD(7)..FOR T=1..SO T-1 IS A FACTOR IN Z7
P=(T-1)(4T^3+6T^2+5T+2)...WE FIND THAT FURTHER NEITHER T=0 IS A FACTOR OR 3T+1 IS A FACTOR...SINCE P(0)=5 AND P(2)=8+2+3+1+5=5MOD(7).WE CHECKED FOR P(2)SINCE{3T+1=0=7...OR....T=2}
HENCE GCD =T-1
HENCE THE GENERATOR FOR THIS IDEAL IS
GENERATOR=H(T)(T-1)+K(T)(T-1)
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