# Algebra Prelim

September 18, 2007

Do as many of the eight problems as you can.
Four completely correct solutions will be a pass; a few complete solutions will count more than many partial solutions.
Always carefully justify your answers.
If you skip a step or omit some details in a proof, point out the gap and, if possible, indicate what would be required to fill it in.

The letters k and K always denote fields.

1. Let K be a field of characteristic zero and L a Galois extension of K. Let f be an irreducible polynomial in K[x] of degree 7 and suppose f has no zeroes in L. Show that f is irreducible in $L [ x ]$

2. Let K be a field of characteristic zero and $f \in K [ x ]$ an irreducible polynomial of degree n. Let L be a splitting field for f. Let G be the group of automorphisms of L which act trivially on K.

(a) Show that G embeds in the symmetric group $S _ { n }$

(b) For each n, give an example of a field K and polynomial f such that $G = S _ { n }$

(c) What are the possible groups G when $n = 3 ?$ Justify your answer.

3. Show there are exactly two groups of order 21 up to isomorphism.

4. (a) Show that the ring Z[i] of Gaussian integers is a unique factorisation domain (UFD).

(b) Is $\mathbb { Z } [ { \sqrt { - 5 } } ]$ a UFD? Justify your answer.

5. Let A be a domain and K its field of fractions.
   Recall that we say $f \in K$ is integral over A if it satisfies an equation

$$
f ^ { n } + a _ { n - 1 } f ^ { n - 1 } + \cdots + a _ { 1 } f + a _ { 0 } = 0 ,
$$

where $a _ { n - 1 } , \ldots , a _ { 0 } \in A$ . The integral closure $\tilde { A } \subset K$ of A is the set of $f \in K$ which are integral over A, and we say A is integrally closed if ${ \tilde { A } } = A$

(a) Show that a UFD is integrally closed.
(Hint: write f as a fraction.)

(b) Compute the integral closure of $k [ x , y ] / ( x ^ { 2 } - y ^ { 3 } )$ . (Remember that a polynomial ring is a UFD and therefore integrally closed.)

(c) Compute the integral closure of $k [ x , y , z ] / ( x ^ { 2 } - y ^ { 2 } z )$ . (Hint: there is an obvious integral element.)

6. Let V be a finite dimensional vector space over Q and $A \colon V \to V$ a linear map such that $A ^ { 7 } = \mathrm { i d }$ , the identity map.
   Suppose that 1 is not an eigenvalue of A. Prove that dim V is divisible by 6.

7. Let V be a vector space over a field k that is not of characteristic two.
   Let ω : $V \times V \to k$ be a non-degenerate skew-symmetric bilinear form, i.e., $\omega ( x , y ) = - \omega ( y , x )$ for all $x , y \in V$ , and if $x \neq 0$ there is a y such that $\boldsymbol { \omega } ( x , y ) \neq \boldsymbol { 0 }$

(a) Show that there exists a basis $e _ { 1 } , \ldots , e _ { n } , f _ { 1 } , \ldots , f _ { n }$ of V such that $\omega ( e _ { i } , e _ { j } ) = \omega ( f _ { i } , f _ { j } ) = 0$ and $\omega ( e _ { i } , f _ { j } ) = \delta _ { i j }$ for all $i , j$ . (In particular dim $V = 2 n$ is even.)

(b) We say a subspace $W \subset V$ is isotropic if $\omega ( w _ { 1 } , w _ { 2 } ) = 0$ for all $w _ { 1 } , w _ { 2 } \in W$ . Show that the dimension of an isotropic subspace is at most 1 dim V .

8. Let H be the ring of quaternions with standard basis 1, i, j, k and identify C with the subring $\mathbb { R } + \mathbb { R } i$ of H.

(a) Use the action of H on itself by left multiplication to explain why there is a ring homomorphism $\varphi : \mathbb { H } \to M _ { 2 } ( \mathbb { C } )$ , where $M _ { 2 } ( \mathbb { C } )$ denotes the ring of $2 \times 2$ matrices.
(Warning: there are two ways to view H as a C-vector space, through right and left multiplication by elements in the subring $\mathbb { R } + \mathbb { R } i . )$

(b) Say why $\varphi$ is injective.

(c) The special unitary group ${ \mathrm { S U } } ( 2 )$ consists of all $2 \times 2$ complex matrices u such that det $( u ) = 1$ and $u u ^ { * } = u ^ { * } u = 1$ where $u ^ { * }$ is the conjugate transpose, i.e., the transpose of the matrix whose entries are the complex conjugates of the entries in $u .$ Show that $\varphi$ restricts to an isomorphism between the group of unit quaternions (those of length one) and SU(2).

(d) Use this to prove that SU(2) acts transitively on the Riemann sphere $\mathbb { C P } ^ { 1 }$ defined as the 1-dimensional subspaces in $\mathbb { C } ^ { 2 }$ . (Hint: use the action of $M _ { 2 } ( \mathbb { C } )$ on $\mathbb { C } ^ { 2 }$ by left multiplication.)

(e) Let $U ( 1 )$ denote the image in ${ \mathrm { S U } } ( 2 )$ of the multiplicative subgroup of $\mathbb { C } - \{ 0 \}$ consisting of the complex numbers $z \in \mathbb { C } \subset$ H of length one.
Show that the coset space $\mathrm { S U } ( 2 ) / U ( 1 )$ is isomorphic to the 2-sphere $S ^ { 2 }$

Remark.
The solution to this problem gives a realization of the Hopf fibration $S ^ { 3 }  S ^ { 2 }$ with fibers $S ^ { 1 }$ because the group of unit quaternions is isomorphic to the 3-sphere $S ^ { 3 }$