Polynomial rings and their automorphisms Vipul Naik A crash course in ring theory The polynomial Polynomial rings and their automorphisms ring Automorphisms and endomorphisms The notions of Vipul Naik invariant subring Some questions about the invariant subring More invariant April 23, 2007 subrings Further connections A summary Polynomial rings Outline and their automorphisms A crash course in ring theory Vipul Naik Definition of ring Modules over rings Generating sets and bases A crash course in Rings and ideals ring theory Concept of subring Definition of ring Modules over rings The polynomial ring Generating sets and The polynomial ring in one variable bases The polynomial ring in many variables Rings and ideals Concept of subring Automorphisms and endomorphisms The polynomial Homomorphism of rings ring Homomorphisms from the polynomial ring Linear and affine endomorphisms Automorphisms and The notions of invariant subring endomorphisms The fixed-point relationship The notions of Some questions about the invariant subring invariant subring Representations and faithful representations Generating sets and questions Some questions about the invariant More invariant subrings subring The orthogonal group Relation between invariant polynomials and vanishing sets More invariant subrings Further connections The module of covariants Further Harmonic polynomials and the Laplacian connections A summary A summary I (R, ∗) forms a semigroup (that is, ∗ is an associative binary operation) I The following distributivity laws hold: a ∗ (b + c) = (a ∗ b) + (a ∗ c) (a + b) ∗ c = (a ∗ c) + (b ∗ c) The identity element for addition is denoted as 0 and the inverse operation is denoted by the prefix unary −. Polynomial rings What’s a ring and their automorphisms Vipul Naik A ring is a set R equipped with two binary operations + A crash course in ring theory (addition) and ∗ (multiplication) such that: Definition of ring Modules over rings Generating sets and I (R, +) forms an Abelian group bases Rings and ideals Concept of subring The polynomial ring Automorphisms and endomorphisms The notions of invariant subring Some questions about the invariant subring More invariant subrings Further connections A summary I The following distributivity laws hold: a ∗ (b + c) = (a ∗ b) + (a ∗ c) (a + b) ∗ c = (a ∗ c) + (b ∗ c) The identity element for addition is denoted as 0 and the inverse operation is denoted by the prefix unary −. Polynomial rings What’s a ring and their automorphisms Vipul Naik A ring is a set R equipped with two binary operations + A crash course in ring theory (addition) and ∗ (multiplication) such that: Definition of ring Modules over rings Generating sets and I (R, +) forms an Abelian group bases Rings and ideals I (R, ∗) forms a semigroup (that is, ∗ is an associative Concept of subring The polynomial binary operation) ring Automorphisms and endomorphisms The notions of invariant subring Some questions about the invariant subring More invariant subrings Further connections A summary Polynomial rings What’s a ring and their automorphisms Vipul Naik A ring is a set R equipped with two binary operations + A crash course in ring theory (addition) and ∗ (multiplication) such that: Definition of ring Modules over rings Generating sets and I (R, +) forms an Abelian group bases Rings and ideals I (R, ∗) forms a semigroup (that is, ∗ is an associative Concept of subring The polynomial binary operation) ring The following distributivity laws hold: Automorphisms I and endomorphisms The notions of a ∗ (b + c) = (a ∗ b) + (a ∗ c) invariant subring Some questions about the invariant (a + b) ∗ c = (a ∗ c) + (b ∗ c) subring More invariant The identity element for addition is denoted as 0 and the subrings inverse operation is denoted by the prefix unary −. Further connections A summary In other words, the multiplication operation is a monoid operation. Note that in any ring, a ∗ 0 = 0 for all a. Hence, a ring with 1 = 0 must be the trivial ring. Polynomial rings Ring with identity and their automorphisms Vipul Naik A crash course in ring theory A ring with identity is a ring for which the multiplication Definition of ring Modules over rings Generating sets and operation has an identity element, that is, there exists an bases Rings and ideals element 1 ∈ R such that: Concept of subring The polynomial a ∗ 1 = 1 ∗ a = a ∀ a ∈ R ring Automorphisms and endomorphisms The notions of invariant subring Some questions about the invariant subring More invariant subrings Further connections A summary Note that in any ring, a ∗ 0 = 0 for all a. Hence, a ring with 1 = 0 must be the trivial ring. Polynomial rings Ring with identity and their automorphisms Vipul Naik A crash course in ring theory A ring with identity is a ring for which the multiplication Definition of ring Modules over rings Generating sets and operation has an identity element, that is, there exists an bases Rings and ideals element 1 ∈ R such that: Concept of subring The polynomial a ∗ 1 = 1 ∗ a = a ∀ a ∈ R ring Automorphisms and endomorphisms In other words, the multiplication operation is a monoid The notions of invariant subring operation. Some questions about the invariant subring More invariant subrings Further connections A summary Polynomial rings Ring with identity and their automorphisms Vipul Naik A crash course in ring theory A ring with identity is a ring for which the multiplication Definition of ring Modules over rings Generating sets and operation has an identity element, that is, there exists an bases Rings and ideals element 1 ∈ R such that: Concept of subring The polynomial a ∗ 1 = 1 ∗ a = a ∀ a ∈ R ring Automorphisms and endomorphisms In other words, the multiplication operation is a monoid The notions of invariant subring operation. Some questions Note that in any ring, a ∗ 0 = 0 for all a. Hence, a ring with about the invariant subring 1 = 0 must be the trivial ring. More invariant subrings Further connections A summary All the rings we shall be looking at today are so-called commutative rings with identity, viz ∗ is commutative and also has an identity element. Polynomial rings Commutative ring and their automorphisms Vipul Naik A crash course in ring theory Definition of ring Modules over rings Generating sets and bases Rings and ideals A ring is said to be commutative if the multiplicative Concept of subring operation ∗ is commutative. The polynomial ring Automorphisms and endomorphisms The notions of invariant subring Some questions about the invariant subring More invariant subrings Further connections A summary Polynomial rings Commutative ring and their automorphisms Vipul Naik A crash course in ring theory Definition of ring Modules over rings Generating sets and bases Rings and ideals A ring is said to be commutative if the multiplicative Concept of subring operation ∗ is commutative. The polynomial ring All the rings we shall be looking at today are so-called Automorphisms and commutative rings with identity, viz ∗ is commutative and endomorphisms also has an identity element. The notions of invariant subring Some questions about the invariant subring More invariant subrings Further connections A summary I We assume multiplication takes higher precedence over addition. This helps us leave out a number of parentheses. For instance, (a ∗ b) + (c ∗ d) can be written simply as ab + cd I Parentheses are also dropped from repeated addition I We denote by n ∈ N the number 1 + 1 + 1 ... 1 where we add 1 to itself n times. Moreover, we denote by nx the number x + x + ... + x (even when there doesn’t exist any 1) The first of these conventions is justified by associativity of multiplication, the second one is justified by the distributivity law. Polynomial rings Conventions followed while writing expressions in and their automorphisms a ring Vipul Naik We generally adopt the following conventions: A crash course in ring theory I The multiplication symbol, as well as parentheses for Definition of ring Modules over rings multiplication, are usually omitted. Thus, a ∗ (b ∗ c) Generating sets and bases may be simply written as abc Rings and ideals Concept of subring The polynomial ring Automorphisms and endomorphisms The notions of invariant subring Some questions about the invariant subring More invariant subrings Further connections A summary I Parentheses are also dropped from repeated addition I We denote by n ∈ N the number 1 + 1 + 1 ... 1 where we add 1 to itself n times. Moreover, we denote by nx the number x + x + ... + x (even when there doesn’t exist any 1) The first of these conventions is justified by associativity of multiplication, the second one is justified by the distributivity law. Polynomial rings Conventions followed while writing expressions in and their automorphisms a ring Vipul Naik We generally adopt the following conventions: A crash course in ring theory I The multiplication symbol, as well as parentheses for Definition of ring Modules over rings multiplication, are usually omitted. Thus, a ∗ (b ∗ c) Generating sets and bases may be simply written as abc Rings and ideals We assume multiplication takes higher precedence over Concept of subring I The polynomial addition. This helps us leave out a number of ring Automorphisms parentheses. For instance, (a ∗ b) + (c ∗ d) can be and written simply as ab + cd endomorphisms The notions of invariant subring Some questions about the invariant subring More invariant subrings Further connections A summary I We denote by n ∈ N the number 1 + 1 + 1 ... 1 where we add 1 to itself n times. Moreover, we denote by nx the number x + x + ... + x (even when there doesn’t exist any 1) The first of these conventions is justified by associativity of multiplication, the second one is justified by the distributivity law. Polynomial rings Conventions followed while writing expressions in and their automorphisms a ring Vipul Naik We generally adopt the following conventions: A crash course in ring theory I The multiplication symbol, as well as parentheses for Definition of ring Modules over rings multiplication, are usually omitted.