Let . Find the Galois group of the splitting field of f(x) over the field of rational numbers. (Deadline April 14.)

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#
Month: March 2017

# Bonus Problem

# HW11, due Wed 4/5/17

# HW10, due Wed 3/29

# No office hours, 3/21-3/31

# HW9, due Wed Mar 22

# HW8, due Friday March 17

# Schedule, Monday March 20-Friday March 31

# HW 7, due Wednesday March 25, 2017

# Some material on Sage/Gap

Let . Find the Galois group of the splitting field of f(x) over the field of rational numbers. (Deadline April 14.)

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- D&F, Section 13.3 #1, #2
- D&F, Section 13.4 #1, #2, #5, #6

- D&F, Section 13.1, #1, #3
- D&F, Section 13.2, #4, #7, #11a, #16, #21

There will be no office hours 3/21-3/31. Bill Rau is available in the MAC at the usual times: 10-12, 4-5 Mondays.

- Let F(A) and F'(A) both satisfy the universal property of a free group on a set A. Show that there is a unique isomorphism from F(A) to F'(A) that is compatible with the canonical maps A -> F(A) and A -> F'(A). Hint: consider the universal property for all possible combinations, taking both F(A) and F'(A) to be the free group and the supplementary group.
- Can you see how a similar argument could be used to show quite generally that objects that satisfy universal properties are unique up to isomorphism?
- DF 5.4 #2,
- DF 5.4 #7,
- DF 5.5 #23-a

Do the following calculations in Sage (not GAP). You may use Sage Cell for the calculations and cut and paste the answers that you submit. You are welcome to improve the code.

- compute 77+31 (no semicolon!)
- compute the set of divisors of 60: divisors(60)
- compute the greatest common divisor: gcd(33,770)
- compute 17 mod 3: mod(17,3)
- compute a and b such that a*64+b*33 = 1, using xgcd(64,33) and check your answer in Sage
- find factor(2^126-1)
- check the largest factor is prime: is_prime(77158673929)
- compute the square of the permutation
- G = SymmetricGroup(7)
*a = G(“(1,2,3,4)(6,5,7)”)*- a^2

- compute the
**“left to right” (warning!)**product of the cycles- G = SymmetricGroup(3)
*a = G(“(1,2,3)”)**b= G(“(2,3)”)**a*b*

- make a list of elements of the permutation group generated by (1,2,3,4,5) and (2,5)(3,4):
*G = SymmetricGroup(5)**g1 = G(“(1,2,3,4,5)”)**g2 =G(“(2,5)(3,4)”)**H = G.subgroup([g1,g2])**H.list()*

- Calculate the conjugacy classes of A5:
*G = AlternatingGroup(5)**G.conjugacy_classes_representatives()*

- Find the derived subgroup of A4:
- G=AlternatingGroup(4)
- G.commutator()

- Calculate a 2-Sylow subgroup of S4:
*SymmetricGroup(4).sylow_subgroup(2)* - Show that the subgroup of S4 generated by (1,2,3) and (2,3,4) is A4:
*G=SymmetricGroup(4)**g1=G(“(1,2,3)”)**g2=G(“(2,3,4)”)**H=G.subgroup([g1,g2])**H.is_isomorphic(AlternatingGroup(4))*

- Calculate a composition series for S4:
*SymmetricGroup(4).composition_series()*

- Monday 3/20, Hales, review of rings, ideals, fields, and extensions
- Wednesday 3/22, Fedorov, D&F Section 13.1, basic theory of field extensions
- Friday 3/24, Fedorov, D&F Section 13.2, algebraic extensions
- Monday 3/27, Xu, D&F, Section 13.3, classical straightedge and compass constructions
- Wednesday 3/29, Yarosh, D&F, Section 13.4 splitting fields
- Friday 3/29, Yarosh, D&F, Section 13.4, algebraic closures

Perform the following calculations in GAP. You can use Sage Cell and choose GAP in the language menu at the bottom-right-hand corner of the input cell. You can cut and paste your answers into the submitted homework.

- compute 77+31;
- compute the set of divisors of 60: DivisorsInt(60);
- compute the greatest common divisor Gcd(33,770);
- compute 17 mod 3;
- compute a and b such that a*64+b*33 = 1, using GcdRepresentation(64,33); check your answer in gap
- find Factors(2^126-1);
- check the largest factor is prime: IsPrime(77158673929);
- compute the square (a^2;) of the permutation
*a:=(1,2,3,4)(6,5,7);* - compute the
**“left to right” (warning!)**product of the cycles*a:= (1,2,3); b:=(2,3); a*b;* - make a list of representatives of the permutation group generated by (1,2,3,4,5) and (2,5)(3,4):
*g:= Group((1,2,3,4,5),(2,5)(3,4)); c:= ConjugacyClasses(g);* - Calculate the conjugacy classes of A5:
*g:= AlternatingGroup(5);*

*c:= ConjugacyClasses(g);* - Find the derived subgroup of A4:
*DerivedSubgroup(AlternatingGroup(4));* - Calculate the 2-Sylow subgroup of S4:
*SylowSubgroup(SymmetricGroup(4),2);* - Show that the subgroup of S4 generated by (1,2,3) and (2,3,4) is A4:
*StructureDescription(Group((1,2,3),(2,3,4)));* - Calculate a composition series for S4:
*DisplayCompositionSeries(SymmetricGroup(4));*

The easiest way to use Sage is through Sage Cell .

Here are some resources for doing group theory in Sage.

- Here is a primer on group theory in Sage by Beezer.
- Or try this tutorial by Joyner and Kohel

- or this Sagemath doc on group theory