Computable Euclidean Domains Asher M. Kach (Joint Work with Rod Downey and with Paul Ellis and Reed Solomon) Southern Wisconsin Logic Colloquium 9 October 2012 Asher M. Kach Computable Euclidean Domains 9 October 2012 1 / 43 Outline 1 The Division Algorithm, Euclid’s Algorithm, and Euclidean Domains 2 Transfinite Euclidean Domains and Rings 3 Effective Algebra: Computing φ and φR Subrings of Q Subrings of K (Xi )i2I 4 Reverse Mathematics 5 Open Questions Asher M. Kach Computable Euclidean Domains 9 October 2012 2 / 43 The Division Algorithm... Problem Divide 18 into 218 (over Z). Answer. Perform long division 12 18 218 180 38 36 2 and so 218 = 12 · 18 + 2. Asher M. Kach Computable Euclidean Domains 9 October 2012 3 / 43 The Division Algorithm... Problem Divide x + 2 into x3 + 18x2 + 2x + 18 (over Q). Answer. Perform long division x2 + 16x − 30 x + 2 x3 + 18x2 + 2x + 18 − x3 − 2x2 16x2 + 2x − 16x2 − 32x − 30x + 18 30x + 60 78 and so x3 + 18x2 + 2x + 18 = (x2 + 16x − 30)(x + 2) + 78. Asher M. Kach Computable Euclidean Domains 9 October 2012 4 / 43 Euclid’s Algorithm... Proposition The algorithm function gcd(a,b) if (a < b) swap(a,b) if (b == 0) return a return gcd(a-b,b) computes the greatest common divisor of nonnegative integers a and b. Problem Find the greatest common divisor of 18 and 10. Answer. Note gcd(18; 10) = gcd(8; 10) = gcd(10; 8) = gcd(2; 8) = gcd(8; 2) = gcd(6; 2) = gcd(4; 2) = gcd(2; 2) = gcd(0; 2) = gcd(2; 0) = 2. Asher M. Kach Computable Euclidean Domains 9 October 2012 5 / 43 Euclid’s Algorithm... Proposition The algorithm function gcd(a,b) if (a < b) swap(a,b) if (b == 0) return a return gcd(a-b,b) computes the greatest common divisor of nonnegative integers a and b. Problem Find the greatest common divisor of 18 and 10. Answer. Note gcd(18; 10) = gcd(8; 10) = gcd(10; 8) = gcd(2; 8) = gcd(8; 2) = gcd(6; 2) = gcd(4; 2) = gcd(2; 2) = gcd(0; 2) = gcd(2; 0) = 2. Asher M. Kach Computable Euclidean Domains 9 October 2012 5 / 43 The Division Algorithm and Euclid’s Algorithm... Remark In both Z and Q[X], the division algorithm (Euclid’s algorithm) terminates because the dividend (either a or b) decreases in size at every step. Within Z, the size of an integer is its magnitude. Within Q[X], the size of a polynomial is its degree. Generalizing this requirement of remainders decreasing in size yields the (traditional) definition of a Euclidean ring. Asher M. Kach Computable Euclidean Domains 9 October 2012 6 / 43 Defining Euclidean Rings and Domains... Definition A ring R (always commutative with multiplicative identity) is a Euclidean ring if there is a function φ : R0 ! ! (where R0 := R nf0g) satisfying (8d 2 R0)(8a 2 R)(9q 2 R)a − qd = 0 or φ(a − qd) < φ(d): The function φ is termed a Euclidean function for R. Definition A Euclidean ring R is a Euclidean domain if it is also an integral domain (i.e., there are no zero divisors). Remark Note the hidden function (existential) quantifier in the definition. Asher M. Kach Computable Euclidean Domains 9 October 2012 7 / 43 Defining Euclidean Rings and Domains... Definition A ring R (always commutative with multiplicative identity) is a Euclidean ring if there is a function φ : R0 ! ! (where R0 := R nf0g) satisfying (8d 2 R0)(8a 2 R)(9q 2 R)a + qd = 0 or φ(a + qd) < φ(d): The function φ is termed a Euclidean function for R. Definition A Euclidean ring R is a Euclidean domain if it is also an integral domain (i.e., there are no zero divisors). Remark Note the hidden function (existential) quantifier in the definition. Asher M. Kach Computable Euclidean Domains 9 October 2012 7 / 43 Defining Euclidean Rings and Domains... Definition A ring R (always commutative with multiplicative identity) is a Euclidean ring if there is a function φ : R0 ! ! (where R0 := R nf0g) satisfying (8d 2 R0)(8a 2 R)(9q 2 R)a + qd = 0 or φ(a + qd) < φ(d): The function φ is termed a Euclidean function for R. Definition A Euclidean ring R is a Euclidean domain if it is also an integral domain (i.e., there are no zero divisors). Remark Note the hidden function (existential) quantifier in the definition. Asher M. Kach Computable Euclidean Domains 9 October 2012 7 / 43 Defining Euclidean Rings and Domains... Definition A ring R (always commutative with multiplicative identity) is a Euclidean ring if there is a function φ : R0 ! ! (where R0 := R nf0g) satisfying (8d 2 R0)(8a 2 R)(9q 2 R)a + qd = 0 or φ(a + qd) < φ(d): The function φ is termed a Euclidean function for R. Definition A Euclidean ring R is a Euclidean domain if it is also an integral domain (i.e., there are no zero divisors). Remark Note the hidden function (existential) quantifier in the definition. Asher M. Kach Computable Euclidean Domains 9 October 2012 7 / 43 Euclidean Functions for Z... Example The integers Z are a Euclidean domain. Proof. The functions φ1(z) := jzj φ2(z) := blog2 jzjc ( jzj if z 6= 5 φ3(z) := 13 otherwise are Euclidean functions for Z. Note that φ3 serves as an example where the implication x divides y implies φ(x) ≤ φ(y) fails (consider x = 5 and y = 10). Asher M. Kach Computable Euclidean Domains 9 October 2012 8 / 43 Euclidean Functions for Z... Example The integers Z are a Euclidean domain. Proof. The functions φ1(z) := jzj φ2(z) := blog2 jzjc ( jzj if z 6= 5 φ3(z) := 13 otherwise are Euclidean functions for Z. Note that φ3 serves as an example where the implication x divides y implies φ(x) ≤ φ(y) fails (consider x = 5 and y = 10). Asher M. Kach Computable Euclidean Domains 9 October 2012 8 / 43 Euclidean Functions for Z... Example The integers Z are a Euclidean domain. Proof. The functions φ1(z) := jzj φ2(z) := blog2 jzjc ( jzj if z 6= 5 φ3(z) := 13 otherwise are Euclidean functions for Z. Note that φ3 serves as an example where the implication x divides y implies φ(x) ≤ φ(y) fails (consider x = 5 and y = 10). Asher M. Kach Computable Euclidean Domains 9 October 2012 8 / 43 An Alternate Definition... Definition (8d 2 R0)(8a 2 R)(9q 2 R)a + qd = 0 or φ(a + qd) < φ(d): Definition If R is a commutative ring, define a sequence of sets fRngn2! via recursion by 0 Rn := fd 2 R :(8a 2 R)(9q 2 R)[a + qd = 0 or a + qd 2 R<n]g S 0 where R<n := m<n Rm and R := R − f0g. Theorem (Motzkin 1949, Samuel 1971) A ring R (respectively integral domain) is a Euclidean ring 0 S (respectively Euclidean domain) if and only if R = n2! Rn. Asher M. Kach Computable Euclidean Domains 9 October 2012 9 / 43 An Alternate Definition... Definition (8d 2 R0)(8a 2 R)(9q 2 R)a + qd = 0 or φ(a + qd) < φ(d): Definition If R is a commutative ring, define a sequence of sets fRngn2! via recursion by 0 Rn := fd 2 R :(8a 2 R)(9q 2 R)[a + qd = 0 or a + qd 2 R<n]g S 0 where R<n := m<n Rm and R := R − f0g. Theorem (Motzkin 1949, Samuel 1971) A ring R (respectively integral domain) is a Euclidean ring 0 S (respectively Euclidean domain) if and only if R = n2! Rn. Asher M. Kach Computable Euclidean Domains 9 October 2012 9 / 43 An Alternate Definition... Definition 0 Rn := fd 2 R :(8a 2 R)(9q 2 R)[a + dq = 0 or a + dq 2 R<n]g Remark 0 R0 := fd 2 R :(8a 2 R)(9q 2 R)[a + dq = 0]g. Thus R0 is the set of units of R as the existential witness q must be the a element − d . In other words, the set R0 consists of those elements d that exactly divide every other a 2 R. Remark 0 R1 := fd 2 R :(8a 2 R)(9q 2 R)[a + dq = 0 or a + dq 2 R0]g. Thus R1 consists of those elements d that either exactly divide or divide modulo a unit remainder every other a 2 R. Asher M. Kach Computable Euclidean Domains 9 October 2012 10 / 43 An Alternate Definition... Definition 0 Rn := fd 2 R :(8a 2 R)(9q 2 R)[a + dq = 0 or a + dq 2 R<n]g Remark 0 R0 := fd 2 R :(8a 2 R)(9q 2 R)[a + dq = 0]g. Thus R0 is the set of units of R as the existential witness q must be the a element − d . In other words, the set R0 consists of those elements d that exactly divide every other a 2 R. Remark 0 R1 := fd 2 R :(8a 2 R)(9q 2 R)[a + dq = 0 or a + dq 2 R0]g. Thus R1 consists of those elements d that either exactly divide or divide modulo a unit remainder every other a 2 R. Asher M. Kach Computable Euclidean Domains 9 October 2012 10 / 43 An Alternate Definition... Definition 0 Rn := fd 2 R :(8a 2 R)(9q 2 R)[a + dq = 0 or a + dq 2 R<n]g Remark 0 R0 := fd 2 R :(8a 2 R)(9q 2 R)[a + dq = 0]g. Thus R0 is the set of units of R as the existential witness q must be the a element − d .