**2018 ALGEBRA MIDTERM EXAM**

A COURSE BY CHIN-LUNG WANG AT NTU

**1.** Determine the conjugacy classes in S

_{5}

## and the number of elements in each class. Use it to show that the only normal subgroups of S

_{5}

## are 1, A

_{5}

## , S

_{5}

## . **2.** Show that there are no simple groups of order 148 or of order 56.

**3.** Show that (i) a finite abelian group is cyclic ⇐⇒ _{exp G} = | G | and (ii) any finite subgroup of the multiplicative group of a field is cyclic.

**4.** Let R be a ring. Prove that if a ∈ R has more than one right inverse then it has infinitely many. (Hint: if ab = 1, show first that 1 − ba 6= 0.) Give an example such that this fails for monoids.

**5.** Let I be an ideal in a ring R. Show that M

n## ( I ) is an ideal in M

n## ( R ) and every ideal in M

n## ( R ) is in this form.

**6.** Let I

1## , . . . , I

n## be pairwise coprime (two-sided) ideals in a ring R in the sense that I

i## + I

j## = R for i 6= j. Show that (i) R/ (

^{T}

^{n}

_{i}

_{=}

_{1}

## I

i## ) ∼ = _{∏}

^{n}

_{i}

_{=}

_{1}

## R/I

i## and (ii) if moreover that R is commutative then

^{T}

^{n}

_{i}

_{=}

_{1}

## I

_{i}

## = _{∏}

^{n}

_{i}

_{=}

_{1}

## I

_{i}

## .

**7.** Let F be a field with | F | = q. Show that (i) the ring of polynomial functions in r variables over F is isomorphic to F [ x

1## , . . . , x

r## ] / h x

^{q}

_{1}

## − x

1## , . . . , x

^{q}r

## − x

r## i and (ii) every function f : F

^{r}

## → F is a polynomial function.

**8.** *Let D be a Euclidean domain whose function δ satisfies: (i) δ* ( ab ) = *δ* ( a ) *δ*

_{(}

## b ) *and (ii) δ* ( a + b ) ≤ max ( *δ* ( a ) *, δ* ( b )) . Show that either D is a field or D = F [ x ] where F is a field.

***** You may prove the existence of Sylow p-subgroups to supplement one (and only one) problem up to 10 pts. Indicate explicitly the problem number.

****** Bonus: Let A = _{Z} [

_{Z}

^{1}

_{2}

## ( 1 + √

## − 19 )] . Show that (i) the norm N : A → _{N is a} Dedekind–Hasse norm and (ii) A is not a Euclidean domain.

_{N is a}

Date: Time and place: pm 1:20 – 5:20, November 9, 2018 at AMB 101.

Note: (1) each problem is of 15 points (total 120 pts), (2) you may work on each part separately, (3) show your answers/computations/proofs in details.

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