Home Syllabus Schedule Links **Homework** Solutions | | | | # Homework ## Suggested practice on Galois theory **A.** For each of the splitting fields *K* in p227 #6, determine Gal(*K*/*Q*) and the lattice of intermediate fields. **B.** Let *K*=*Q*(√*p*_{1},...,√*p*_{k}). Show that Gal(*K*/*Q*) is isomorphic to (*Z*_{2})^{k}. **C.** If *F* contained in the constructible numbers, and *F* is a finite extension over *Q*, what can you say about the group structure of Gal(*F*/*Q*)? Notation in all: let *Q* denote the rational numbers, √p be the square root of *p*, and *Z*_{2} be the integers mod 2. ## Homework 12, due Apr 12 at 12:07pm Please do the following problems from the textbook: Herstein, p227: 5, 6, 7, 10, 14 Herstein, p231: 1, 2, 3, 4, 5 (note that *F* is any subfield of the reals) Hint on p231 #1: similar triangles may be helpful. ## Homework 11, due Apr 15 at 12:07pm Please do the following problems from the textbook: Herstein, p214: 5, 8, 10 Herstein, p219: 2 Herstein, p227: 11, 12, 13 and the following additional problems: **A.** If *K* is an algebraic field extension of *F*, give the details of the induction proof that *F*(*a*_{1}, *a*_{2}, ..., *a*_{n}) is a finite degree extension of *F* (where each *a*_{i} is in *K*). **B.** Let *f* and *g* be polynomials in *F*[*x*], and that *f* divides *g* in some *K*[*x*] for some extension field *K* of *F*. Show that *f* divides *g* in *F*[*x*]. **C.** Let *f* be an irreducible polynomial in *F*[*x*] of degree *p*. Suppose *K* is a algebraic field extension of *F* such that *f* is not irreducible in *K*[*x*]. Show that *p* divides [*K*:*F*]. Hint: Consider an extension field *L* of *K* such that *f* has a root in *L*[*x*]. **D.** (More challenging) Show that if *a* and *b* are elements of a field extension *K* over *F*, then *a* is algebraic over *F*(*b*) if and only if *b* is algebraic over *F*(*a*). Hint: Look at the extension field *F*(*a*,*b*) over *F*(*a*). ## Homework 10, due Apr 8 at 12:07pm Please do the following problems from the textbook: Herstein, p167: 1, 2 Herstein, p214: 3, 4, 9 and the following additional problems: **A.** Show that the face ring of any simplicial complex (over *F*) is an *F*-algebra. ## Homework 10, due Apr 3 (Friday) at 12:07pm Please do the following problems from the textbook: Herstein, p161: 3, 4, 5 Herstein, p167: 6, 16, 17 ## Homework 9, due Mar 25 at 12:07pm Please do the following problems from the textbook: Herstein, p142: 5, 6 Herstein, p149: 3, 7, 8 Herstein, p152: 1, 2, 3, 8 Herstein, p158: 1, 2, 3, 7 Herstein, p166: 8 ## Homework 8, due Mar 18 at 12:07pm Please do the following problems from the textbook: Herstein, p130: 4, 6, 7, 8 Herstein, p135: 5, 6, 7, 9, 10, 12 Herstein, p142: 2, 3, 4 and the following additional problems: **A.** Explain why the ring in p135 #12 is not a field or division ring. **B.** ~~Exercise 1~~Exercise 6 from the Zorn's Lemma and Maximal Ideals handout. ## Homework 7, due Mar 4 at 12:07pm Please do the following problems from the textbook: Herstein, p102: 17 Herstein, p108: 4, 5, 6, 14, 16 Herstein, p116: 11, 13 Herstein, p130: 9, 10 (Hint: On p102 #17, remember that normal Sylow subgroups are characteristic.) and the following additional problems: **A.** Show that there are no nonabelian simple groups of order < 101, with the exception of the simple group of order 60, and possibly orders 90 and 96. **B.** (More challenging) Show that there are no simple groups of order 90 or 96. ## Homework 6, due Feb 25 at 12:07pm Please do the following problems from the textbook: Herstein, p80: 9, 10, 11, 21a-c Herstein, p90: 3, 9, 10 Herstein, p102: 7, 12a-b, 13 (Hint: On p90 #9, the Sylow C-Theorem may save you some work!) and the following additional problems: **A.** Explain why *p*-groups cannot be simple and nonabelian. **B.** Explain why no groups of order < 24 are nonabelian simple. ## Homework 5, due Feb _**20**_ at 12:07pm Please do the following problems from the textbook: Herstein, p80: 2, 3, 4, 5, 8a Herstein, p90: 2 Actions handout: Exercise 21, Exercise 37 and the following additional problems: **A.** Using your result from p90 #2, find a normal subgroup of order 4 in *S*_{4}. **B.** What is the kernel of the conjugation action of *G* on *G*? **C.** Show that *S*_{6} has a subgroup of index 6. (Hint: use an action!) **D.** Let *G* be a simple group of order 360. Show that *G* has no subgroups of index 2, 3, 4, or 5. (You might be interested to know that *G* is a subgroup of *S*_{6}, and has a subgroup of index 6 for a reason similar to **C**.) ## Homework 4, due Feb 11 at 12:07pm Please do the following problems from the textbook: Herstein, p65: 15 Herstein, p70: 1, 4, 5, 7, 8, 9, 21 Herstein, p75: 6, 7 Herstein, p90: 8 (Use Lemma 13 from the handout) and the following additional problems: **A.** Let *G* be a finite group, and *T* be a homomorphism from *G* to *K*, and let *H* be a subgroup of *G*. Show that [*T*(*G*) : *T*(*H*)] divides [*G* : *H* ], and |*T*(*H*)| divides |*H*|. **B.** For what groups *G* is the map *I* defined by *I*(*g*)=*g*^{-1} an automorphism? **C.** p70, problem 14. Hint: there are two classes of groups that you may want to handle as different cases. ## Homework 3, due Feb 4 at 12:07pm Please do the following problems from the textbook: Herstein, p53: 4, 5, 12, 13, 16 Herstein, p65: 1, 5, 11, 16, 18, 19 and the following additional problems: **A.** Let *G* be finite and abelian, and let *m* divide |*G*|. Show that *G* has a subgroup of order *m*. Hint: Apply Cauchy's Theorem for Abelian Groups inductively. ## Homework 2, due Jan 28 at 12:07pm Please do the following problems from the textbook: Herstein, p46: 2, 3, 6, 8, 10, 11, 14 Herstein, p53: 1, 2, 10, 11 (Hint on p46 #2: Assume *G* has an element of infinite order...) and the following additional problems: **A.** Let *D*_{10} be the dihedral group with 10 elements, that is, all rotations and flips of the pentagon. Describe all subgroups of *D*_{10}. Which subgroups are normal? * ***B.** Let *H*, *K*, and *L* be subgroups of *G*, with *H*⊆*K*. Prove that *K*∩*HL*=*H*(*K*∩*L*).
*Homework 1, due Jan 21 at 12:07pm*
* Please do the following problems from the textbook:* Herstein, p35: 1, 3, 7, 9, 10, 12, 13, 24 Herstein, p46: 1, 4 and the following additional problems: **A.** In Herstein p35 #24, the resulting group *G* is of order 6. *S*_{3} is also of order 6. Compare and contrast *G* with *S*_{3}. **B.** Suppose that *G*=*H*∪*K* where *H* and *K* are subgroups of *G*. Show that either *H*=*G* or *K*=*G*. (Red text added later.) |