Summary of lectures

The following will be updated as we go with information about what we covered on particular days. References are to Artin's book, but we will not cover topics in the same order.

• 1/6/25 11.1 Definition of ring (possibly allowing noncommutative multiplication). Examples. Basic properties of the 0 element. Units in a ring. Rings will be commutative from now on unless otherwise stated. Fields, including the example of Q(sqrt(2)). Subrings.
• 1/8/25 11.2 Zero-divisors and integral domains. The 0 ring. A a field is an integral domain. The polynomial ring R[x] and degree of elements. If R is an integral domain then so is R[x]. Evaluation of polynomials. Example of polynomial long division.
• 1/10/25 No class, professor out of town

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• 1/13/25 11.2 Proof of polynomial long division: for f,g with g monic there is unique q, r with f = qg + r, where r is 0 or has degree less than g. The remainder and factor theorems.
• 1/15/25 11.3 If R is an integral domain than a polynomial of degree n in R[x] has at most n roots in R. Homomorphisms of rings. The map R[x] -> R given by evaluation at some fixed a is a homomorphism. Ideals. Kernels of homomorphisms are ideals and images are subrings. Definition of the factor ring R/I for an ideal I.
• 1/17/25 11.3 A ring is a field if and only if it has exactly two ideals (0) and R. The ideals in Z are (0) and all nZ for positive n. The ideal (a_1, ... , a_n) generated by a subset. Principal ideals (a) and principal ideal domains. Z is a PID. If F is a field, a nonzero ideal I in F[x] is principal and equal to (g) where g is a monic poly of lowest degree among nonzero elements in I. The kernel of evaluation F[x] to F given by f(x) -> f(a) is the ideal (x-a). One can evaluate into a ring containing the coefficient ring. The evaluation map R[x] to C given by f(x) -> f(i) has kernel (x^2 + 1).

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• 1/20/25  MLK holiday, no class
• 1/22/25 11.4 The 1st isomorphism theorem for rings:  if phi: R to S is a homomorphism, then there is an isomorphism from R/(ker phi) to phi(R).  Examples:  F[x]/(x-a) is isomorphic to F for a field F.  R[x]/(x^2 +1) is isomorphic to C.  Q(sqrt(2)) is isomorphic to Q[x]/(x^2 -2).  Ideal correspondence theorem:  The ideals of a factor ring R/I are in one-to-one correspondence with the ideals J of R containing I--- explicitly if J is such an ideal then J/I= {x + I | x in J} is the corresponding ideal of R/I.   Theorem on factor rings of factor rings: if I is contained in J then (R/I)/(J/I) is isomorphic to R/J.  Examples.
• 1/24/25 11.2, 11.8  Prime ideals.  An ideal I is prime if and only if R/I is a domain.  Examples of prime ideals.  The ring of polynomials R[x_1, ... x_n] in n variables over a ring R.  Theorem:  this ring can be considered as an iterated polynomial extension R[x_1][x_2}... [x_n] (proof omitted).  (y) is a prime ideal of F[x,y], and F[x,y]/(y) is isomorphic to F[x].  Maximal ideals.  An ideal I is maximal if and only if R/I is a field.  Maximal ideals are prime.   The maximal ideals of the ring Z of integers are the ideals (p) generated by prime numbers.  The prime ideals of Z are those maximal ideals together with (0) which is prime but not maximal. 

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• 1/27/25:  Products of rings.  The ring R x S consists of all ordered pairs of elements (r, s) with coordinate-wise operations.  Chinese remainder theorem:  Z/mnZ is isomorphic to Z/mZ x Z/nZ when m and n are relatively prime.  Q[x]/(x^2-4) is isomorphic to Q x Q.  An element e is idempotent if e^2 = e.  If a ring R has a pair of idempotents e_1, e_2 such that e_1 + e_2 = 1 and e_1e_2 = 0, then R is isomorphic to S x T where S = e_1R and T = e_2R are rings with identity elements e_1 and e_2 respectively.  Products of finitely many rings can be characterized similarly.  Example:  rings in which every element is idempotent are called Boolean and the finite ones are isomorphic to (Z/2Z)^n for some n (full proof not given).
• 1/29/25 Fields of fractions.  Given an integral domain R, the field of fractions F is a new ring with elements equivalence classes of fractions r/s where s is not 0, where r/s is equivalent to t/u if ru = ts, and with standard operations of addition and multiplication.  Proof that F is a field containing R (sketch).  The field of rational functions F(x) for a field F is the field of fractions of F[x].  The universal property of the field of fractions.  Example, Q(sqrt(2)) is the field of fractions of Z[sqrt(2)].
• 1/31/25 irreducible elements in a domain.  Irreducible polynomials in F[x], F a field.  Polynomials of degree 1 are irreducible and polynomials of degree 2 or 3 are irreducible if and only if they have no roots in F.  Prime elements in domains, associate elements.  In a domain, a and b are associate if and only if (a) = (b), and a nonzero element p is prime if and only if (p) is a prime ideal.   Main theorem:  If R is a PID, then for a nonzero nonunit element a, (a) is maximal if and only if a is irreducible if and only if a is prime.

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• 2/3/2025 The prime spectrum of a PID.  Definition of a UFD.   Theorem:  A PID is a UFD.  Lemma:  given an ascending chain of ideals in a ring, the  union is an ideal, and if the ring is a PID then the chain stabilizes.  Proof of the theorem:  in a PID, if a nonzero nonunit element is not a product of irreducibles, then inductively we can construct a properly ascending chain of ideals, contradicting the lemma.  For uniqueness, one uses that irreducible elements are prime in a PID and induction.
• 2/5/2025 Euclidean Domains. Z and F[x] for a field F are Euclidean domains. A Euclidean domain is a PID. Review of complex numbers. The Gaussian integers Z[i] is a Euclidean domain with respect to the function d(a + bi) = a^2 + b^2 (the square of the usual complex norm). Example of how the idea of the proof can be used to actually find q, r such that f = qg + r with d(r) < d(g), for given f, g in Z[i].
• 2/7/2025 Rings of the form Z[\sqrt{d]] for integers d such that \sqrt{d} is not rational. The norm function N(a + b \sqrt{d}) = | a^2 - db^2| is multiplicative. y is a unit if and only if N(y) = 1, and if N(y) is a prime integer then y is irreducible. If d < -1 then the units are only +-1, and if d = -1 the roots are +-1 and +-i. Z[\sqrt{2}] has an infinite group of units. Z[\sqrt{-6]] is not a UFD, because 6 = (-2)(3) = (\sqrt(-6))(\sqrt(-6)) and all elements are irreducible and non-associate. Theorem: in a UFD, irreducible and prime elements are the same. This often gives a shorter way of verifying a ring is not a UFD, by showing an irreducible is not prime.

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• 2/10/2025 Factorization in Z[i]. If p is a prime number in Z, then either p is also irreducible in Z[i] or else is it reducible and is a product of precisely two irreducibles (a + bi) and (a-bi). The latter case occurs if and only if p is expressible as a sum of 2 squares. Theorem: The primes of the first types are precisely the primes that are congruent to 3 mod 4; the prime 2 and primes congruent to 1 mod 4 are of the latter type. Lemma used in the proof: The multiplicative group of nonzero elements in a finite field is a cyclic group (proof deferred). The irreducibles in Z[i] are precisely those of the 2 types above. Statement of Gauss' Theorem characterizing integers that are a sum of 2 squares (proof deferred). Example: how to find a and b such that a^2 + b^2 = (2)(3^2)(5)(13).
• 2/12/2025 Midterm Exam
• 2/14/2025 Finishing up Factorization in Z[i]. Proof of the theorem characterizing integers that are a sum of 2 squares. Z[i] is isomorphic to Z[x]/(x^2 + 1), and so for a prime p, Z[i]/(p) is isomorphic to Z[x]/(x^2 +1, p), which is isomorphic to (Z/pZ)[x]/(x^2 + 1). When p is congruent to 3 mod 4, this is a field with p^2 elements, while when p is congruent to 1 mod 4 or p = 2 then writing p = (a + bi)(a-bi), we have Z[i]/(a + bi) is isomorphic to Z/pZ. Sketch of proof of theorem that the multiplicative group of a finite field is cyclic. We assume without proof a key lemma from group theory: a finite group of order n such that for all divisors d of n there are at most d elements of order dividing n is a cylic group.

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• 2/17/2025 President's Day Holiday (no class)
• 2/19/2025 More on irreducible polynomials. Statement of the fundamental theorem of algebra. Monic irreducibles over C[x] are just the linear polynomials x-a. Monic irreducibles in R[x] are the linear polynomials x-a and the quadratic polynomials x^2 + a_1x + a_0 with no real roots (in which case they have a pair of conjugate complex roots). To find irreducibles in (Z/pZ)[x] for a prime p, one can work inductively. Example: showing a polynomial of degree 4 over a finite field is irreducible by showing it has no roots and is not a product of two degree 2 irreducibles (first find all degree 2 irreducibles). The rational root theorem for testing if a polynomial in Q[x] has a root in Q, which allows one in theory to decide if any degree 2 or 3 polynomial is irreducible over Q.
• 2/21/2025 Factorization in Z[x] versus factorization in Q[x]. A polynomial a_0 + a_1 x + ... + a_n x^n in Z[x] is called primitive if gcd(a_0, a_1, ..., a_n) = 1. A prime p in Z is irreducible in Z[x]. A polynomial f(x) in Z[x] of degree at least 1 which is primitive and irreducible in Q[x] is irreducible in Z[x]. Crucial lemma: a product of 2 primitive polynomials in Z[x] is again primitive (proof used reduction mod a prime). Gauss's Lemma: if h in Z[x] has a factorization h = f g with f,g in Q[x], then there is a scalar r in Q such that g' = rg and f' = r^{-1} f are in Z[x], so h = f' g' has a factorization in Z[x].

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• 2/24/2025 Eisenstein criterion for irreducibility: Given a polynomial f= a_0 + a_1 x + ... + a_n x^n in Z[x] and a prime p such that a_n is not a multiple of p; a_i is a multiple of p for 0 <= i <= n-1, and a_0 is not a multiple of p^2, then f is irreducible in Q[x]. The proof uses the reduction mod p homomorphism. Example: x^p -1 = (x-1) Phi(x) where Phi(x) = 1 + x + x^2 + ... + x^{p-1}. Using a substitution we can show that Phi(x+1) satisfies the Eisenstein criterion for the prime p, so it is irreducible over Q[x]. Then Phi(x) is also irreducible over Q[x], so also over Z[x] since it is primitive. The irreducibles in Z[x] are precisely the primes in Z and the primitive polynomials of degree at least 1 that are irreducible in Q[x]. Proof that Z[x] is a UFD.
• 2/26/2025 GCD's. Definition of a gcd(a,b) for a, b not both 0 in a domain R. If d is gcd of a and b then so is any associate of d, so gcd's are defined only up to associates. In a UFD, any pair of elements not both 0 has a gcd, which can be computed from the factorizations of the elements as products of irreducibles. If R is a PID, then gcd(a, b) is equal to the generator d of the principal ideal such that (a, b) = (d). In particular, d = ax + by for some x, y in R in case R is a PID. In a Euclidean domain, there is Euclidean algorithm which can be used to compute gcd(a, b). In a general UFD, gcd(a,b) may not be equal to an R-linear combination of a and b. Example: gcd(2, x) = 1 in Z[x] but (2,x) is not a principal ideal and 1 is not equal to 2f + gx for polynomials f and g. Theorem: If R is a UFD then R[x] is a UFD. Proof is omitted, but it is very similar to the proof that Z[x] is a UFD. In particular it mainly relies on the fact that R has gcd's and that F[x] is a UFD where F is the field of fractions of R. Review of examples of rings we have studied which are Euclidean, PIDs, UFDS, or not.
• 2/28/2025 Class cancelled

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• 3/3/2025 Vector spaces V over an arbitrary field F. Definition. F^n. If R is a ring containing F as a subring then we can think of R as an F-vector space. Def of what is means for a subset S of V to be linearly independent over F, or to span V. A basis is a subset that spans and is independent. A basis of F^n is given by the n standard basis vectors e_i. The polynomial ring F[x] is an F-vector space with basis of all powers of x: {1, x, x^2, ....}. C is an R-vector space with basis {1, i}. Finite and infinite dimensional vector spaces. In a finite dimensional vector space, all bases have the same number of elements, called the dimension (proof on HW). If f(x) is an irreducible monic polynomial in F[x] of degree n, then E = F[x]/(f) is a field containing F and it is an F-vector space of dimension n. An explicit basis of E as an F-vector space is given by the elements 1 + (f), x+(f), ...., x^{n-1} + (f).
• 3/5/2025 Posets. Maximal elements. Totally ordered posets. A chain is a totally ordered subset of a poset. Upper bounds. Zorn's Lemma: If S is a nonempty poset and every chain in S has an upper bound in S, then S contains a maximal element. Zorn's Lemma is equivalent to the axiom of choice (no proof). Use of Zorn's Lemma to prove that every vector space has a basis, by letting S be the collection of all linearly independent subsets, which is a poset under inclusion of subsets. Example: R has a basis over Q, though it is not possible to write down any basis explicitly.
• 3/7/2025 Linear Transformations of vector spaces over a field. Example: phi: C to C given by complex conjugation is a linear transformation as vector spaces over R, but not over C. Subspaces. Factor vector spaces V/W. Kernels and images of linear transformations are subspaces. 1st isomorphism theorem for vector spaces. Example: projection onto a subspace. For V finite dimensional, dim(V/W) = dim V - dim W. Rank/nullity Theorem for linear transformations.

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• 3/10/2025 let phi: V to V where V is finite dimensional with basis B = {v_1, ... v_n}. The matrix of the linear transformation with respect to B is M_B(phi) = B = (b_{ij}) where phi(v_j) = sum_i b_{ij} v_i. If a vector v is thought of as a column vector a where v = sum_i a_i v_i, then writing phi(v) = sum_i c_i v_i we have c = Ba. If phi and psi are both linear maps V to V then M_B(psi compose phi) is equal to the matrix product M_B(psi) M_B(phi). Invertible matrices. B and C are similar if there exists an invertible P such that C = P^{-1} BP. If C = {w_1, ... w_n} is another basis then M_C(phi) = P^{-1} M_B(phi) P, where P is the change of basis matrix such that w_j = \sum_i p_{ij} v_i. Diagonalizable linear transformations and matrices. Eigenvectors. A linear transformation phi: V to V is diagonalizable if and only if V has a basis consisting of eigenvectors of phi. An example of a linear transformation which is not diagonalizable.