I : Newstead definitions Basic mob of unity in a study as - - PowerPoint PPT Presentation

SLIDE 1

Roots of

Unity

I

:

Basics of

Roots of

Unity

SLIDE 2

Newstead

① Basic definitions

② study

mob of unity

in a

③ roots of unity

as subgroups of

F

#

SLIDE 3

Define If

F

is

a field , then The

roots of

X

"

• I

c- FIX]

are

called

the

nth root of unity

in F

.

Nate

F

has at

most

n

nth

nots of unity

EI

IR

has

relatively few

roots of unity

2nd

roots

i

.

{ l ,

• l )

even" roots:{ I,

• t)
• ddth

root of unity

:

113

SLIDE 4

why do

we

care about

nots of

unity

?

① Ipa

• GF lpn)

is the

splitting field

• f

XP

"

• x overIp

⇐ Gffpn)

is the splitting field for

xp

"-t

• I
• ver Ip

so :

Epn

= GF(pm

,

is just th

collection of

Cpa

• Dst

root of

unity ( together

with

O)

• ver Ip

Hence :

we

create

finite fields

by adjoining

roots

• f

unity

to

Kp

SLIDE 5

② we've

also

studied

splitting field for

x'-2e

Ex)

extensively

( we

called

it

E)

.

we've

seen

that if

a

is

some

root of

x3-2

, Then

⑨ K) t E

.

However

:

theorem ( Reds of unity

and

binomials ) Suppose that

xn - I

C- FIX)

splits

in

F

, then if

KIF

contains

• ne

root

x

• f

x

"

• f

c- Flex] ,

Then

we

get

xn -f

splits

in

K

.

SLIDE 6

Uts

'

analyze

roots of unity

in a

• let

ne IN

.

Then The

set

{ (e

"%)

"

.. ask an }

= {

cos12¥ )ti sm (HI)

: ""}

is

a

set

• f

n

dishnot

solutions to

x

"

• I =D over Cl .

( ie , this is the set

• f

nth

root of unity

in ¢)

( this

comes

up

in the

next

problem

set

.)

SLIDE 7

To

what degree

is the

behavior et mob of unity

in

¢

carried

• ver

to

a general

field ?

In

a general

field ,

are

the

nth

roots of

unity

a

cyclic

subgroup

• f

F

# ?yes !

Theorem ( Finite

subgroups of F # are cyclic)

IFGE F #

and

IGKA , then

G

is cyclic

.

( this

is Theorem

62

in

text)

SLIDE 8

PI

let

GE

F#

with

n

• IGI sis

.

We

want :

G

is

cyclic

.

Recall

from

the

Fund

. Thm .

• f Finite Ab. Groups

we

have

G

= Km, ④

• - ⑦Ink

.

Observe

:

n

• IGI

=

m ,

.

• - Mk .

Recall

also :

G

is

cyclic

iff

{mi ,

. - -ink)

are

pairwise relatively

prime ⇐

1cm Em, ,

Mk)

• n

.

Note that

we already

know

n

• M ,
• -Mk slam { mi ,

.

mis)

SLIDE 9

If

we

write

m

• 1cm { m , ,

Mk )

.

Then for

any g EG

,

we

get

a

corresponding (g , ,

• y ga )

c- Km

,

④

• - -④ Kink

.

Lagrange

says

Igi l l mi

⇒ lgillm

for

all

i

.

Hence

we

get

Ilg . .

. . ., gal I

m

,

and

heave

lgllm

.

Hence

gm

• I

for

all

GEG

.

So

all

• g. EG

are

roots of

XM - I

.

Hence

NHGIEM

1¥

,

SLIDE 10

tar If

F

is

a field

and

NEIN

, then

the nth

roots of

unity

in

F term

a

cyclic

subgroup

• f

F #

.