3 Polynomials and Unique Factorisation

Question 1

Defintion 3.1:
Fields

Answer 1

A field is a set 𝔽 with operations of addition and multiplication.
i.e. for every 𝑥,𝑦∈𝔽, then 𝑥+𝑦 ∈ 𝔽 and 𝑥⋅𝑦 = 𝑥𝑦 ∈ 𝔽,
such that the field axioms hold.

Question 2

Definition 3.1:
Field axioms (F1)-(F4)

Addition behaves well

Answer 2

(F1) + is associative: ∀𝑥,𝑦,𝑧∈𝔽, (𝑥+𝑦)+𝑧=𝑥+(𝑦+𝑧)
(F2) There is an additive identity 0∈𝔽 s.t. 0+𝑥 = 𝑥 = 𝑥+0 ∀𝑥∈𝔽
(F3) Every 𝑥∈𝔽 has an additive inverse −𝑥∈𝔽 such that 𝑥+(−𝑥) = 0 = −𝑥+𝑥
(F4) + is commutative: ∀𝑥,𝑦∈𝔽 𝑥+𝑦 = 𝑦+𝑥.

Question 3

Definition 3.1:
Field axioms (F5)

Addition and multiplication are compatiable

Answer 3

(F5) ⋅ is distributive over +: ∀𝑎,𝑥,𝑦∈𝔽
𝑎⋅(𝑥+𝑦) = 𝑎⋅𝑥 + 𝑎⋅𝑦
(𝑥+𝑦)⋅𝑎 = 𝑥⋅𝑎 +𝑦 ⋅𝑎.

Question 4

Definition 3.1:
Field axioms (F6)-(F9)

Multiplication behaves well

Answer 4

(F6) ⋅ is associative: ∀𝑥,𝑦,𝑧∈𝔽 (𝑥⋅𝑦)⋅𝑧 = 𝑥⋅(𝑦⋅𝑧)
(F7) There is a multiplicative identity 1∈𝔽 s.t. 1⋅𝑥 = 𝑥 = 𝑥⋅1 ∀𝑥∈𝔽
(F8) ⋅ is commutative: ∀𝑥,𝑦∈𝔽 𝑥⋅𝑦 = 𝑦⋅𝑥
(F9) 1≠0 and ∀𝑥∈𝔽∖{0} there is a multiplicative inverse 𝑥^(−1) s.t. 𝑥^(−1)⋅𝑥 = 1 = 𝑥⋅𝑥^(−1).

Question 5

Definition 3.1:
Extended field axioms

Answer 5

(F3x) ∀𝑥∈𝔽, we have −(−𝑥)=𝑥
(F5x) ∀𝑎,𝑥∈𝔽
𝑎⋅0 = 0 = 0⋅𝑎
𝑎⋅(−𝑥) = −(𝑎⋅𝑥)
(−𝑥)⋅𝑎 = −(𝑥⋅𝑎)
(F9x) ∀𝑥∈𝔽∖{0}, [𝑥^(−1)]^(−1) = 𝑥.

Question 6

Remark 3.2:
Additive groups, rings and commutative rings

Answer 6

𝔽 (with +,0) is an additive group if (F1)-(f4) holds.
𝔽 is a ring if (F1)-(F7) holds.
𝔽 is a commutative ring if (F1)-(F8) holds.

Question 7

Definition 3.4:
Integral domain

Answer 7

A commutative ring 𝔽 is an integral domain if, ∀𝑥,𝑦∈𝔽, we have that 𝑥𝑦=0 ⟹ 𝑥=0 or 𝑦=0.

Question 8

Lemma 3.5:
Cancellation law

Answer 8

A commutative ring 𝔽 is an integral domain if and only if ∀𝑥,𝑦,𝑧∈𝔽, we have that
( 𝑥𝑦=𝑥𝑧 and 𝑥≠0 ) ⟹ 𝑥=𝑧.

Question 9

Definition 3.7:
Polynomials

Answer 9

A polynomial in 𝑋 with coefficients in 𝔽 is an expression 𝑓 = 𝑓(𝑋) = ∑ₙ₌₀∞ 𝑎ₙ𝑋ⁿ, where the coefficients 𝑎ₙ∈𝔽, and where all but finitely many of the 𝑎ₙ are zero.

Two polynomials are equal if and only if all of their coefficients are equal.

We write 𝔽[𝑋] for the resulting polynomial ring with coefficients in 𝔽.

Question 10

Definition 3.7:
Addition and multiplication of polynomials

Answer 10

𝑓 = ∑ₙ₌₀∞ 𝑎ₙ𝑋ⁿ,
𝑔 = ∑ₙ₌₀∞ 𝑏ₙ𝑋ⁿ, then

𝑓+𝑔 = ∑ₙ₌₀∞ (𝑎ₙ+𝑏ₙ)𝑋ⁿ
𝑓⋅𝑔 = ∑ₙ₌₀∞ (∑ⁿₖ₌₀ 𝑎ₖ𝑏ₙ₋ₖ)𝑋ⁿ.