Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN: 9781285463230
Author: Gilbert, Linda, Jimmie
Publisher: Cengage Learning,
Question
[画像: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 (r)=2- (b) This remains true when Q is replaced with any extension field F, where QCFCC. eat b√√2-c√3+d√ a+bv2+ c√] + dvb a b√2+c√√3 d√√6 a+b√2-c√3+0√6 a+b√2-cv3-dv6 a+b√2-c√√√√6a-b√2-c√] + dvb They form the Galois group of x 5x +6. The multiplication table and Cayley graph cre 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. Problem 19: Galois Groups of Degree 6 Extensions Let f(x)=-3-2x2+x+1€ Q[x]. . Find the Galois group of the splitting field of f(x) over Q. Remarks x=√2+√3 is a primitive element of F. ie. Q(a) = Q(√2√3). ■しかくThere is a group action of Gal(f(x)) on the set of roots 5 = (±√2±√3) of f(x). • How does the structure of the Galois group reflect the symmetries of the roots? An example: the Galois correspondence for f(x) = x3-2 Consider Q(C. 2)=Q(a), the splitting field of f(x)=x3-2. QKC) It is also the splitting field of m(x)=x+108, the minimal polynomial of a=√2√-3. Q2) (2) (3) Q(C) Q(2) Q(32) Q(<23/2) Let's see which of its intermediate subfields are normal extensions of Q. Q: Trivially normal. Q(C. √2) ■しかくQ(C): Splitting field of x2+x+1; roots are C.(2 = Q(C). Normal. ■しかくQ(V2): Contains only one root of x3-2, not the other two. Not normal. ■しかくQ(C2): Contains only one root of x3-2, not the other two. Not normal. ■しかくQ(2): Contains only one root of x3-2, not the other two. Not normal. ■しかくQ(C. V2): Splitting field of x3-2. Normal. By the normal extension theorem, Gal(Q(C)) (Q(C): Q1=2, Gal(Q(C. 2)) = [Q(C. 2): Q = 6. Moreover, you can check that | Gal(Q(2)) =1<[0(2): Q] = 3. QKC. 2) 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). ■しかく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). ■しかく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).]
expand button
Transcribed Image Text: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 (r)=2- (b) This remains true when Q is replaced with any extension field F, where QCFCC. eat b√√2-c√3+d√ a+bv2+ c√] + dvb a b√2+c√√3 d√√6 a+b√2-c√3+0√6 a+b√2-cv3-dv6 a+b√2-c√√√√6a-b√2-c√] + dvb They form the Galois group of x 5x +6. The multiplication table and Cayley graph cre 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. Problem 19: Galois Groups of Degree 6 Extensions Let f(x)=-3-2x2+x+1€ Q[x]. . Find the Galois group of the splitting field of f(x) over Q. Remarks x=√2+√3 is a primitive element of F. ie. Q(a) = Q(√2√3). ■しかくThere is a group action of Gal(f(x)) on the set of roots 5 = (±√2±√3) of f(x). • How does the structure of the Galois group reflect the symmetries of the roots? An example: the Galois correspondence for f(x) = x3-2 Consider Q(C. 2)=Q(a), the splitting field of f(x)=x3-2. QKC) It is also the splitting field of m(x)=x+108, the minimal polynomial of a=√2√-3. Q2) (2) (3) Q(C) Q(2) Q(32) Q(<23/2) Let's see which of its intermediate subfields are normal extensions of Q. Q: Trivially normal. Q(C. √2) ■しかくQ(C): Splitting field of x2+x+1; roots are C.(2 = Q(C). Normal. ■しかくQ(V2): Contains only one root of x3-2, not the other two. Not normal. ■しかくQ(C2): Contains only one root of x3-2, not the other two. Not normal. ■しかくQ(2): Contains only one root of x3-2, not the other two. Not normal. ■しかくQ(C. V2): Splitting field of x3-2. Normal. By the normal extension theorem, Gal(Q(C)) (Q(C): Q1=2, Gal(Q(C. 2)) = [Q(C. 2): Q = 6. Moreover, you can check that | Gal(Q(2)) =1<[0(2): Q] = 3. QKC. 2) 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). ■しかく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). ■しかく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).
Expert Solution
Check Mark
Knowledge Booster
Background pattern image
Recommended textbooks for you
Text book image
Elements Of Modern Algebra
Algebra
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Cengage Learning,
Text book image
Algebra and Trigonometry (MindTap Course List)
Algebra
ISBN:9781305071742
Author:James Stewart, Lothar Redlin, Saleem Watson
Publisher:Cengage Learning
Text book image
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:9781133382119
Author:Swokowski
Publisher:Cengage
Text book image
Linear Algebra: A Modern Introduction
Algebra
ISBN:9781285463247
Author:David Poole
Publisher:Cengage Learning