Instruction: Do not use AI. : Do not just give outline, Give complete solution with visualizations. : Handwritten is preferred. The "One orbit theorem" Let n and r be roots of an irreducible polynomial over Q. Then (a) There is an isomorphism : Q(n) Q(2) that fixes Q and with (n) = 2. (b) This remains true when Q is replaced with any extension field F, where QCFCC. e: a+b+c+da+b√2+ c√3+ √ a+b√2-c√3+0√6 a+b√2-c√3-dv6 a+b√√2-ca-b√2-c√√3+ d√б They form the Galois group of x 5x +6. The multiplication table and Cayley graph are shown below. 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. Remarks a=√2+v3s 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 = (±√2 ±√3) of f(x). Problem 8: Galois Group of a Quintic Polynomial Let f(x) = 2-2-1 € Q[z]. • Determine the Galois group of the splitting field of f (a) over Q. • Is the Galois group solvable? Justify your answer. An example: the Galois correspondence for f(x) = x³-2 Consider Q(C. 2) Q(a), the splitting field of f(x)=x-2. It is also the splitting field of m(x)=x+108, the minimal polynomial of a = √√√-3. Let's see which of its intermediate subfields are normal extensions of Q. ■ Q: Trivially normal. Q(2) Q(2) Q²) 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(2): 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. 2): 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<[Q(32): Q] = 3. Q(2) Q(2) Q((2/2) c. &2) Subfield lattice of Q(C. 32) = D (125) Subgroup lattice of Gal(Q(C. 2)) = D ■The automorphisms that fix Q are precisely those in Ds. ■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. 2). The normal subgroups of D3 are: D3. (r) and (e).