Extend The Automorphism E E We Will Show Has Order 4 E Show 0 1 2 Thus 1 2 Without Loss Of Generality 1 (17.98 KiB) Viewed 42 times
Extend The Automorphism E E We Will Show Has Order 4 E Show 0 1 2 Thus 1 2 Without Loss Of Generality 2 (13.14 KiB) Viewed 42 times
Extend The Automorphism E E We Will Show Has Order 4 E Show 0 1 2 Thus 1 2 Without Loss Of Generality 3 (13.14 KiB) Viewed 42 times
Extend The Automorphism E E We Will Show Has Order 4 E Show 0 1 2 Thus 1 2 Without Loss Of Generality 4 (29.8 KiB) Viewed 42 times
Extend the automorphism E-E. We will show has order 4 (e) Show (0)-(-1+√2²@ Thus, ()=(-1+√2). Without loss of generality, let a(0)=(-1+√2). (1) Show ¹)-8. (x) Verify a(6)-(-1+√20. Thus, has onder 4 in E. Similarly, it can be shown that has order in E, where (0)-(3-√3) (h) Use the above information to show or(@)-0(0) 7 of 8 The eight roots of f(x) are ±0, 105, 2₂, where 4-√√(2+√√2)(3+√3) ₁-√√(2+√2)(3-√3) 0₂-√√(2-√2)(3+√3) -√√(2-√2)(3-√3) (1) A portion of the table of automorphisms of E that fix Q has been filled in below. Com plete the rest of the table. ior---r √2 √2 √3 √3 #0₂0₂0₂-0₁
Extend the automorphism o: E→ E. We will show has order 4: (e) Show (0²) = (-1+ √2)² 0². Thus, 7(0) = (-1+√2). Without loss of generality, let (0) = (-1+ √2) 0. (f) Show 2(0) = -0. (g) Verify (0)=-(-1+ √2)0.
Thus, has order 4 in E. Similarly, it can be shown that 7 has order 4 in E, where 7(0) =(3-√3)0. (h) Use the above information to show 07(0) = -70 (0). 7 of 8 The eight roots of f(x) are 10₁, 102, 103, 104 where 0₁ (2+ √2)(3+√3) 0₂=√(2-√2)(3+√3) 03 = √(2+√2)(3-√3) 0₁ (2-√2)(3-√3) (i) A portion of the table of automorphisms of E that fix Q has been filled in below. Com- plete the rest of the table. -7-07 √2 √2 √3 √3 0₁ 01 0₂ 03 04-0₁
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