Arithmetic of polynomials

Learning tasks

A) Divisibility in \(\mathbb{K}[X]\)
I) Recall: divisibility relation
1) Applying divisibility properties	Ex 1
II) Associated polynomials
2) Recognizing associated polynomials and using the monic form	Ex 2 Ex 3 Ex 4
B) GCD and Euclid's algorithm
I) Definition of the gcd
3) Reading the monic gcd from a factorisation	Ex 5
II) Fundamental idea and Euclid's algorithm
4) Computing \(A \wedge B\) by Euclid's algorithm	Ex 6 Ex 7 Ex 8 Ex 9
III) Common divisors and the gcd
5) Listing common divisors of two polynomials	Ex 10
IV) Bézout's identity
6) Computing Bézout coefficients in \(\mathbb{K}[X]\)	Ex 11 Ex 12 Ex 13
C) Least common multiple
I) Definition
7) Reading the monic lcm from a factorisation	Ex 14
II) Common multiples
8) Listing common multiples of two polynomials	Ex 15
III) The gcd-lcm relation
9) Computing \(A \vee B\) and using the gcd-lcm relation	Ex 16 Ex 17 Ex 18
IV) Worked example
10) Computing gcd and lcm by factorisation	Ex 19
D) Coprime polynomials
I) Definition and Bézout's theorem
11) Recognizing coprime polynomials	Ex 20 Ex 21 Ex 22
II) Gauss's lemma and consequences
12) Applying Bézout's theorem and Gauss's lemma	Ex 23 Ex 24 Ex 25 Ex 26
III) Application: coprimality via common complex roots
13) Deciding coprimality via complex roots	Ex 27
E) Arithmetic with several polynomials
I) Definition and recursive computation
14) Computing gcd of several polynomials	Ex 28
II) Bézout for several polynomials
15) Computing a Bézout combination for three polynomials	Ex 29 Ex 30
III) Coprime as a set vs.~pairwise coprime
16) Distinguishing coprime as a set from pairwise	Ex 31 Ex 32
F) Going further
I) Synthesis problems	Ex 33 Ex 34 Ex 35

Lesson
Text book

Exercises Correction

Convention

Throughout this exercise sheet, \(\mathbb{K} \in \{\mathbb{R}, \mathbb{C}\}\) and a divisor is always a non-zero polynomial (course convention). Notations: \(A \wedge B\) for the monic gcd, \(A \vee B\) for the monic lcm, \(\mathbb{K}[X]^*\) for the non-zero polynomials, \(\sim\) for the association relation. The gcd \(A \wedge B\) is defined when \((A, B) \neq (0, 0)\); the lcm \(A \vee B\) is defined for \(A, B \in \mathbb{K}[X]^*\) and extended by convention \(A \vee B := 0\) when \(A = 0\) or \(B = 0\).

A) Divisibility in \(\mathbb{K}[X]\)
    I) Recall: divisibility relation
      1) Applying divisibility properties Ex 1
    II) Associated polynomials
      2) Recognizing associated polynomials and using the monic form Ex 2 Ex 3 Ex 4
B) GCD and Euclid's algorithm
    I) Definition of the gcd
      3) Reading the monic gcd from a factorisation Ex 5
    II) Fundamental idea and Euclid's algorithm
      4) Computing \(A \wedge B\) by Euclid's algorithm Ex 6 Ex 7 Ex 8 Ex 9
    III) Common divisors and the gcd
      5) Listing common divisors of two polynomials Ex 10
    IV) Bézout's identity
      6) Computing Bézout coefficients in \(\mathbb{K}[X]\) Ex 11 Ex 12 Ex 13
C) Least common multiple
    I) Definition
      7) Reading the monic lcm from a factorisation Ex 14
    II) Common multiples
      8) Listing common multiples of two polynomials Ex 15
    III) The gcd-lcm relation
      9) Computing \(A \vee B\) and using the gcd-lcm relation Ex 16 Ex 17 Ex 18
    IV) Worked example
      10) Computing gcd and lcm by factorisation Ex 19
D) Coprime polynomials
    I) Definition and Bézout's theorem
      11) Recognizing coprime polynomials Ex 20 Ex 21 Ex 22
    II) Gauss's lemma and consequences
      12) Applying Bézout's theorem and Gauss's lemma Ex 23 Ex 24 Ex 25 Ex 26
    III) Application: coprimality via common complex roots
      13) Deciding coprimality via complex roots Ex 27
E) Arithmetic with several polynomials
    I) Definition and recursive computation
      14) Computing gcd of several polynomials Ex 28
    II) Bézout for several polynomials
      15) Computing a Bézout combination for three polynomials Ex 29 Ex 30
    III) Coprime as a set vs.~pairwise coprime
      16) Distinguishing coprime as a set from pairwise Ex 31 Ex 32
F) Going further
    I) Synthesis problems Ex 33 Ex 34 Ex 35

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