MATH 120B Quiz: Math_120B_-_Quiz_7_Ver._1_(Fall17)SOLUTIONS
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In this quiz you may use both sides of this sheet for your work if you need to. [k : q] = hk : q( 3 2)i hq( 3 2) : qi hq( 3) : qi = 2 divides [k : q] hq( 3 2) : qi = 3 divides [k : q] = 6 divides [k : q] (since lcm(2, 3) = Now notice that hk : q( 3)i 3 because 3 2 is a root of x3 2 q( 3)[x] = irr( 3 2, q( 3)) divides x3 2 (in particular irr is of lower degree). We conclude (b) compute k : q( 3) : we know that a tower of elds gives. 6 [k : q] 6 = [k : q] = 6. [k : q] = hk : q( 3)i hq( 3) : qi. = 6 = hk : q( 3)i 2.
