Instruction: Do not use AI. : Do not just give outline, Give complete solution with visualizations. : Handwritten is preferred. The "One orbit theorem" Let and be roots of an irreducible polynomial over Q. Then (a) There is an isomorphism : Q(n)→Q(2) that foxes Q and with (n) = 2. (b) This remains true when Q is replaced with any extension field F, where QCFCC. e: a+b+c√3+d√ a+bv2+ c√] + d√b a+b√2-c√3+0√6 a+b√√2-c√√] + √ a b√√2+c√√ √ a+b√2-c√3-d√6 a-b√2-c√√3+d√б Fundamental theorem of Galois theory Given f€ Z[x], let F be the splitting field of f. and G the Galois group. Then the following hold: (a) The subgroup lattice of G is identical to the subfield lattice of F, but upside-down. Moreover, HG if and only if the corresponding subfield is a normal extension of Q. (b) Given an intermediate field QC KCF, the corresponding subgroup H< G contains precisely those automorphisms that fix K. They form the Galois group of x 5x +6. The multiplication table and Cayley graph are shown below. Remarks x=√2+√3 is a primitive element of F, ie, Q(a) = Q(√2√3). ■There is a group action of Gal(f(x)) on the set of roots 5= S=(±√2.±√3) of f(x). Consider Q(C. 2)= Q(a), the splitting field of f(x)=x³-2. It is also the splitting field of m(x)=x+108, the minimal polynomial of Q(2) Q(2) Q(2) Let's see which of its intermediate subfields are normal extensions of Q. ■Q: Trivially normal. Q(C. √2) Problem 19: Galois Groups of Degree 6 Extensions Let f(x)=x -3- 2x²+x+1€ Q[x]. ⚫ Find the Galois group of the splitting field of f(x) over Q. • How does the structure of the Galois group reflect the symmetries of the roots? An example: the Galois correspondence for f(x) = x³-2 Q(C) Q(32) Q(32) Q(<23/2) Q(C. √2) ■Q(C): Splitting field of x²+x+1; roots are C.(² = Q(C). Normal. ■Q(V2): Contains only one root of x³-2, not the other two. Not normal. ■Q(C2): Contains only one root of x3-2, not the other two. Not normal. ■ Q(2): Contains only one root of x3-2, not the other two. Not normal. ■Q(C. V2): Splitting field of x³-2. Normal. By the normal extension theorem, | Gal(Q(C)) = [Q(C): Q]=2, Gal(Q(C. 2))| = [Q(C. 2): Q = 6. Moreover, you can check that | Gal(Q(2)) =1<[0(2): Q] = 3. Subfield lattice of Q(C. 32) = Dr Subgroup lattice of Gal(Q(C. 2)) = D The automorphisms that fix Q are precisely those in D3. The automorphisms that fix Q(C) are precisely those in (r). The automorphisms that fix Q(2) are precisely those in (f). The automorphisms that fix Q(C2) are precisely those in (rf). ■The automorphisms that fix Q(22) are precisely those in (2). The automorphisms that fix Q(C. 2) are precisely those in (e). The normal field extensions of Q are: Q. Q(C), and Q(C. V/2). The normal subgroups of D3 are: D3. (r) and (e).