Aulas I Mm445

❆♥é✐s ❝♦♠✉t❛t✐✈♦s✱
❊✉❝❧✐❞❡❛♥♦s✱ ❢❛t♦r✐❛✐s✱ ♣r✐♥❝✐♣❛✐s ❡ ◆♦❡t❤❡r✐❛♥♦s

P♦❞❡♠♦s ❞✐③❡r q✉❡ ❡♠ ❣❡r❛❧ ♦s ❛♥é✐s sã♦ ♣❛rt❡s ❞❡ ✉♠ ❝♦r♣♦ ♦♥❞❡ ♥ã♦ t❡♠♦s ❞✐✈✐sã♦✳ ❆ ♣❛❧❛✈r❛
❛♥❡❧ ❞❡✈❡ t❡r✲s❡ ♦r✐❣✐♥❛❞♦ q✉❛♥❞♦ ●❛❧♦✐s ❡st✉❞♦✉ ♣❡❧❛ ♣r✐♠❡✐r❛ ✈❡③ ❝♦r♣♦s ✜♥✐t♦s✳ ◆♦ ❡st✉❞♦ ❞❛s
❝♦♥❣r✉ê♥❝✐❛s ♠ó❞✉❧♦

n

♦ q✉❡ ❢❛③❡♠♦s é ✏❡♥r♦❧❛r✑ ♦ ❝♦♥❥✉♥t♦ ❞♦s ✐♥t❡✐r♦s ♥✉♠ ❝ír❝✉❧♦✳ ❊♥r♦❧❛r ❡♠

❝ír❝✉❧♦ ❧❡✈❛ ❛ ❛♥❡❧✳ ❊♥✜♠✱ ❡ss❡ é ♦ ♥♦♠❡✳ P♦r ❡①❡♠♣❧♦✿

Z

❡

F [x]

sã♦ ❛♥é✐s ❝♦♠ ♦s q✉❛✐s ❡st❛♠♦s

❝♦♥✈✐✈❡♥❞♦✳ ▼❛s ♣♦❞❡♠♦s ❢❛❜r✐❝❛r ♠✉✐t♦s ♦✉tr♦s✳ P❛r❛ ✉♠ ❝♦♥❥✉♥t♦ s❡r ❛♥❡❧ ❞❡✈❡ t❡r ✉♠❛ ✉♥✐❞❛❞❡
♠✉❧t✐♣❧✐❝❛t✐✈❛✱ ✐st♦ é✱ ✉♠

1✳

0✱

❉❡✈❡ t❡r ✉♠ ③❡r♦✱

t❡r s♦♠❛ ❡ ♣r♦❞✉t♦ ❞❡ s❡✉s ❡❧❡♠❡♥t♦s✳ ❖ ♣r♦❞✉t♦

Mn

❞❡✈❡ s❡r ❞✐str✐❜✉t✐✈♦ ❡♠ r❡❧❛çã♦ ❛ s♦♠❛✳ P♦r ❡①❡♠♣❧♦✿
♣r♦♣r✐❡❞❛❞❡s✱ t❡♠

1 q✉❡ é ❛ ♠❛tr✐③ ✐❞❡♥t✐❞❛❞❡❀

t❡♠

♦ ❝♦♥❥✉♥t♦ ❞❛s ♠❛tr✐③❡s

0 q✉❡ é ❛ ♠❛tr✐③ ♥✉❧❛✳

n ♣♦r n t❡♠ ❡ss❛s

❙♦♠❛♠♦s ❡ ♠✉❧t✐♣❧✐❝❛♠♦s

♠❛tr✐③❡s ❡ ❛ ♠✉❧t✐♣❧✐❝❛çã♦ é ❞✐str✐❜✉t✐✈❛ ❡♠ r❡❧❛çã♦ ❛ s♦♠❛✳ ▲♦❣♦ é ✉♠ ❛♥❡❧✳ ❆ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ❡ss❡
❛♥❡❧ ❡ ♦s ❞♦✐s ♦✉tr♦s ❡①❡♠♣❧♦s✱

Z ❡ F [x]✱

é q✉❡ ❡♠

Mn

❛ ♠✉❧t✐♣❧✐❝❛çã♦ ♥ã♦ é ❝♦♠✉t❛t✐✈❛✳ ❆♥é✐s ♦♥❞❡

❛ ♠✉❧t✐♣❧✐❝❛çã♦ é ❝♦♠✉t❛t✐✈❛ sã♦ ❝❤❛♠❛❞♦s ❞❡ ❛❜❡❧✐❛♥♦s ❡♠ ❤♦♠❡♥❛❣❡♠ ❛ ❆❜❡❧✳ ❖✉tr♦ ❡①❡♠♣❧♦ ❞❡
❛♥❡❧ ♥ã♦ ❝♦♠✉t❛t✐✈♦ é ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞❛s ❛s ❢✉♥çõ❡s
♣r♦❞✉t♦ ✉s✉❛❧ ❞❡ ❢✉♥çõ❡s✿

(f + g)(t) = f (t) + g(t)

❡

F = {f : R → R | f

❢✉♥çã♦

}

❝♦♠ ❛ s♦♠❛ ❡ ♦

(f g)(t) = f (t)g(t)✳

❖❜s❡r✈❛çã♦✳ ❚♦❞♦ ❝♦r♣♦ é t❛♠❜é♠ ✉♠ ❛♥❡❧ ❡ t♦❞♦ s✉❜❝♦r♣♦ é t❛♠❜é♠ ✉♠ s✉❜❛♥❡❧✳

▼❛s é ❝❧❛r♦

q✉❡ ♥ã♦ ✈❛❧❡ ❛ r❡❝í♣r♦❝❛✳
◆❛ ✈❡r❞❛❞❡ ♣♦❞❡♠♦s ❞✐③❡r q✉❡ ✉♠ ❝♦r♣♦

x

t❡♠ ✉♠ ✐♥✈❡rs♦

x−1

❡♠

F✳

F

é ✉♠ ❛♥❡❧ t❛❧ q✉❡ ♣❛r❛ t♦❞♦

A

x = 0✱

❡♥tã♦

✉♠ ❡❧❡♠❡♥t♦ q✉❡ t❡♠ ✐♥✈❡rs♦ é ❝❤❛♠❛❞♦ ❞❡ ✉♠❛ ✉♥✐❞❛❞❡s✳

❉❡✜♥✐çã♦✳ ❙❡❥❛ A ✉♠ ❛♥❡❧ ❡ A× = { u ∈ A |
A

s❡

◆✉♠ ❛♥❡❧ ❞❡✈❡♠♦s ❡s♣❡r❛r q✉❡ ♠✉✐t♦s ❡❧❡♠❡♥t♦s✱ ❡♠❜♦r❛ ♥ã♦ ♥✉❧♦s✱

♥ã♦ t❡♥❤❛♠ ✐♥✈❡rs♦✳ ❊♠ ✉♠ ❛♥❡❧

❡❧❡♠❡♥t♦s ❞❡

x ∈ F✱

q✉❡ t❡♠ ✐♥✈❡rs♦ ❡♠

❡①✐st❡

v∈A

t❛❧ q✉❡

uv = 1 }✳

■st♦ é

A×

sã♦ ♦s

A✳

❖❜s❡r✈❛çã♦✳ ◆❛ ✈❡r❞❛❞❡ A× é ✉♠ ❣r✉♣♦ ❡♠ r❡❧❛çã♦ ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❞♦ ❛♥❡❧✳
P♦r ❡①❡♠♣❧♦✱ s❡

A

❢♦r ✉♠ ❝♦r♣♦✱ ❡♥tã♦

Z[i]