Exercises for Thursday and Friday
... Important Ideas and Useful Facts: (i) Sets and elements: A set is a collection of objects, referred to as elements. A set may be represented, for example, by a list of elements surrounded by curly brackets and separated by commas, or using set builder notation {. . . | . . .}, where the vertical lin ...
... Important Ideas and Useful Facts: (i) Sets and elements: A set is a collection of objects, referred to as elements. A set may be represented, for example, by a list of elements surrounded by curly brackets and separated by commas, or using set builder notation {. . . | . . .}, where the vertical lin ...
SOLUTIONS TO EXERCISES 1.3, 1.12, 1.14, 1.16 Exercise 1.3: Let
... b) ((i) =⇒ (ii):) Suppose every nonzero element of k is a root of unity. Then every nonzero element satisfies the polynomial xm − 1 for some positive integer m. Let n = (1 + 1 + · · · + 1), the n-fold sum of 1. Then either 2 = 0 (hence k has characteristic 2), or there exists m ∈ N such that 2m = 1. ...
... b) ((i) =⇒ (ii):) Suppose every nonzero element of k is a root of unity. Then every nonzero element satisfies the polynomial xm − 1 for some positive integer m. Let n = (1 + 1 + · · · + 1), the n-fold sum of 1. Then either 2 = 0 (hence k has characteristic 2), or there exists m ∈ N such that 2m = 1. ...
Lecture 12
... which exends R and in which every polynomial equation has a solution. It turns out that to do this, one simply has to add a solution, which is denoted i, of the polynomial x2 + 1 and then extend the algebra of R in a natural way to incorperate this new number. This leads to the field C of complex nu ...
... which exends R and in which every polynomial equation has a solution. It turns out that to do this, one simply has to add a solution, which is denoted i, of the polynomial x2 + 1 and then extend the algebra of R in a natural way to incorperate this new number. This leads to the field C of complex nu ...
PHI 312
... Show that if A has at least 9 elements and the set [A]2 of (unordered) pairs of elements of A is partitioned into two pieces P and Q, then either there is a subset B of A with at least 4 elements such that all pairs in [B]2 belong to P or else there is a subset C of A with at least 3 elements such t ...
... Show that if A has at least 9 elements and the set [A]2 of (unordered) pairs of elements of A is partitioned into two pieces P and Q, then either there is a subset B of A with at least 4 elements such that all pairs in [B]2 belong to P or else there is a subset C of A with at least 3 elements such t ...
Introduction to Coding Theory
... Proof: (Existence) For q = pn , consider xq − x in Zp [x], and let F be its splitting field over Zp . Since its derivative is qxq−1 − 1 = −1 in Zp [x], it can have no common root with xq −x and so, xq −x has q distinct roots in F . Let S = {a ∈ F : aq −a = 0}. Then S is a subfield of F since S conta ...
... Proof: (Existence) For q = pn , consider xq − x in Zp [x], and let F be its splitting field over Zp . Since its derivative is qxq−1 − 1 = −1 in Zp [x], it can have no common root with xq −x and so, xq −x has q distinct roots in F . Let S = {a ∈ F : aq −a = 0}. Then S is a subfield of F since S conta ...
Topological Quantum Field Theories in Topological Recursion
... Not surprisingly the motivation behind the axioms of a Topological Quantum Field Theory comes from Quantum Field Theory. To define a Quantum Field Theory on some manifold we must specify some action, S(φ), which is a function of the fields, φ, on the manifold. This action completely determines the p ...
... Not surprisingly the motivation behind the axioms of a Topological Quantum Field Theory comes from Quantum Field Theory. To define a Quantum Field Theory on some manifold we must specify some action, S(φ), which is a function of the fields, φ, on the manifold. This action completely determines the p ...
First order justification of C = 2πr
... completeness of real closed fields, this theory is also complete4 . 1 These include Pasch’s axiom (B4 of [Har00]) as we axiomatize plane geometry. Hartshorne’s version of Pasch is that any line intersecting one side of triangle must intersect one of the other two. 2 These axioms are equivalent to th ...
... completeness of real closed fields, this theory is also complete4 . 1 These include Pasch’s axiom (B4 of [Har00]) as we axiomatize plane geometry. Hartshorne’s version of Pasch is that any line intersecting one side of triangle must intersect one of the other two. 2 These axioms are equivalent to th ...
I. Existence of Real Numbers
... 6a. Let ∼ be the relation on N × N defined by (a, b) ∼ (c, d) whenever a + d = b + c. Show that ∼ is an equivalence relation on N × N. 6b. Let Z denote the set of equivalence classes of ∼. Show that if (a, b) ∼ (a′ , b′ ) and (c, d) ∼ (c′ , d′ ) then (a+c, b+d) ∼ (a′ +c′ , b′ +d′ ). (Conclude: there ...
... 6a. Let ∼ be the relation on N × N defined by (a, b) ∼ (c, d) whenever a + d = b + c. Show that ∼ is an equivalence relation on N × N. 6b. Let Z denote the set of equivalence classes of ∼. Show that if (a, b) ∼ (a′ , b′ ) and (c, d) ∼ (c′ , d′ ) then (a+c, b+d) ∼ (a′ +c′ , b′ +d′ ). (Conclude: there ...
Sections 2.7/2.8 – Real Numbers/Properties of Real Number
... A real number is any number that belongs to the set of rational numbers or the set of irrational numbers. Each real number corresponds to a point on the number line. Each real number is either negative, zero, or positive. ...
... A real number is any number that belongs to the set of rational numbers or the set of irrational numbers. Each real number corresponds to a point on the number line. Each real number is either negative, zero, or positive. ...
Introduction for the seminar on complex multiplication
... interesting subject for a talk in the seminar. Kronecker’s Jugendtraum. This is not actually an analogue of the map z 7→ exp(2πiz) yet, but it comes very close. It does give A(F )[b] as an analogue of Gm (Q)[n]. Moreover, we may see in a later talk how to use this theorem to express the field CMF,Ψ ...
... interesting subject for a talk in the seminar. Kronecker’s Jugendtraum. This is not actually an analogue of the map z 7→ exp(2πiz) yet, but it comes very close. It does give A(F )[b] as an analogue of Gm (Q)[n]. Moreover, we may see in a later talk how to use this theorem to express the field CMF,Ψ ...
18. Cyclotomic polynomials II
... First, claim that if f (x) is irreducible in some ( /p)[x], then it is irreducible in [x]. A factorization f (x) = g(x)·h(x) in [x] maps, under the natural -algebra homomorphism to ( /p)[x], to the corresponding factorization f (x) = g(x) · h(x) in ( /p)[x]. (There’s little reason to invent a notati ...
... First, claim that if f (x) is irreducible in some ( /p)[x], then it is irreducible in [x]. A factorization f (x) = g(x)·h(x) in [x] maps, under the natural -algebra homomorphism to ( /p)[x], to the corresponding factorization f (x) = g(x) · h(x) in ( /p)[x]. (There’s little reason to invent a notati ...
Study Guide
... or subtracting terms and multiplying terms (x + x is different than x * x) Be able to add or subtract terms and multiply or divide terms Understand the difference between adding terms and combining terms (One can add any two terms together, but one can only COMBINE like terms) Understand the Distrib ...
... or subtracting terms and multiplying terms (x + x is different than x * x) Be able to add or subtract terms and multiply or divide terms Understand the difference between adding terms and combining terms (One can add any two terms together, but one can only COMBINE like terms) Understand the Distrib ...
Math 248A. Norm and trace An interesting application of Galois
... of a minimal polynomial. This concrete viewpoint is how norms and traces arise very often, but the general concept as defined above is a bit more subtle than this and provide a theory with much better properties than such a naive viewpoint would suggest. One important point is that for a higher-degr ...
... of a minimal polynomial. This concrete viewpoint is how norms and traces arise very often, but the general concept as defined above is a bit more subtle than this and provide a theory with much better properties than such a naive viewpoint would suggest. One important point is that for a higher-degr ...
Homomorphism of Semigroups Consider two semigroups (S, ∗) and
... (a) Let M be the set of all 2 × 2 matrices with integer entries. The determinant of any matrix ac bd A= is denoted and defined by det(A) = |A| = ad − bc. One proves in Linear Algebra that the determinant is a multiplicative function, that is, for any matrices A and B , det(AB) = det(A) · det(B) Thus ...
... (a) Let M be the set of all 2 × 2 matrices with integer entries. The determinant of any matrix ac bd A= is denoted and defined by det(A) = |A| = ad − bc. One proves in Linear Algebra that the determinant is a multiplicative function, that is, for any matrices A and B , det(AB) = det(A) · det(B) Thus ...