隨風而去 發問時間: 科學數學 · 1 0 年前

代數 環的問題

第一題 Let a be an element of order 15 in a group

Find 0(a*9) , 0(a*10) , 0(a*8)

解答為 0(a*9) = 5

0(a*10) =3

0(a*8) =15

請問這答案怎麼來的 完全看不懂環的概念 希望有過程

第二題

Find 0(3) and 0(7) in U(Z20) Find out if U(Z20) is a cyclic group

請問這怎麼算 希望有過程

1 個解答

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  • 1 0 年前
    最佳解答

    Question 1 order(a)=15=>a^15=e

    For a^9, we need to find the minimum value of n such that 9n is a multiple of 15. The answer is 5. So order(a^9)=5.Similarly, order(a^10)=3, order(a^8)=10

    Question 2

    The elements in U(Z_20) are 1,3,5,7,9,11,13,17,19

    By Euler's theorem x^ϕ(n) = 1 (mod n). Here n=9, ϕ(n)=4. So, the order of 3 and 7 should be a factor of 4. As 3^1=3,3^2=9,3^4=81=1 (mod 20). order (3)=4. 7^1=7,7^2=49=9, 7^4=2401=1 (mod 20). order(7)=4. U(Z_20) is not a cyclic group. Actually, it is isomorphic to C_2*C_4

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