Lectures:     M-W-F  9:00 AM--10:50 AM    HSS 1315 Office Hour:     M-W-F  11:00 AM--12:30 PM   APM 7230

Discussion: TuTh 10:00 AM--10:50 AM  HSS 1315 TA: Hugo Jenkins (hajenkinsucsd edu) Office hour: TuTh 9 AM--10 AM HSS 5012

Book John B. Fraleigh, A first course in Abstract Algebra 8th edition. (The main text book). Lecture notes will be posted in this webpage. J. A. Gallian, Contemporary abstract algebra.

Schedule This is a tentative schedule for the course. If necessary, it may change.

Assignments. Problem sets will be posted here. Make sure to refresh your bowser. I would like to thank Zach Higgins for providing the solutions to the homework sets. (Solutions will not be necessarily available after the course is finished.) Due August 7: Here is the first problem set. The following problems are not part of the problem set, but you have to know how to solve them for the exams. Section 22, problems 1, 3, 6, 8, 19, 20, 21, 22, 23, 24, 25, 27, 28, 32, 38, 40, 44. Due August 14: Here is the second problem set. The following problems are not part of the problem set, but you have to know how to solve them for the exams. Section 23, problems 2, 3, 10, 12, 14, 30. Section 24, problem 27. Due August 21: Here is the third problem set. The following problems are not part of the problem set, but you have to know how to solve them for the exams. Section 26, problems 1, 2, 17. Section 27, problems 4, 7, 11, 16, 17, 25, 27. Section 28, problems 1, 2, 3, 4, 6, 8, 9, 10, 11, 26, 34, 35, 36. Due August 28: Here is the fourth problem set. The following problems are not part of the problem set, but you have to know how to solve them for the exams. Section 28, problems 12, 14, 16, 17, 18, 19, 21, 22, 30, 31. Section 30, probelms 1, 2, 3, 17, 18, 26, 37. Due September 4: Here is the fifth problem set. The following problems are not part of the problem set, but you have to know how to solve them for the exams. Section 31, problems 1, 2, 3, 4, 5, 6, 8, 15, 16, 17, 18, 19, 24, 30. Section 34, problems 1, 2, 3, 4, 6, 8, 25, 30. Section 39, problems 25, 30, 34.

Summary of my lectures You are supposed to read the relevant sections of your book.

8/4. (1), Chapter 22 In the first lecture, we pointed out that a large part of algebra has been about the study of zeros of polynomials. Then, we defined ring, and commutative and unital rings. We proved that a unital ring has a unique identity. Next, we proved that in every ring \(0\cdot a=a\cdot 0=0\), \((-a)\cdot b=a\cdot (-b)=-(a\cdot b)\), and \((-a)\cdot(-b)=a\cdot b\). We proved the subring criterion. Recalled the subgroup criterion and reprove it. Recalled cyclic subgroups. Recalled that subgroups of a cyclic group are cyclic. Showed that every subgroup of \(\mathbb{Z}\) is a subring. Pointed out that the set of positive integers is closed under addition and multiplication, but it is not a subring. Mentioned that the set of even integers is a commutative ring, but is not unital. Defined the ring \({\rm M}_n(A)\) of \(n\)-by-\(n\) matrixes with entries in \(A\). Pointed out that if \(A\) is unital, so is \({\rm M}_n(A)\) for every \(n\). Mentioned that \({\rm M}_n(A)\) is not commutative if \(A\) is unital and non-trivial and \(n\geq 2\). 8/6. (2), Chapter 22 and 23 In the second lecture, we defined the ring of \(A\)-valued functions on an arbitrary set \(X\). Defined the support of a function \(f:X\to A\), and showed that for every \(f,g\in F(X,A)\), we have \(fg=0\) if the intersection of the support of \(f\) and the support of \(g\) is empty. Next, we mentioned that if \(X=\{1,\ 2,\ \cdots,\ n\}\), then \(F(X,A)\) can be identified with \(A\times \cdots \times A\) \(n\)-times. This was used as a motivation to define the direct product of \(n\) rings. Mentioned that if \(A_1,\ldots,A_n\) are unital rings, then their direct product is unital and the identity of \(A_1\times \cdots \times A_n\) is \( (1_{A_1},\ldots,1_{A_n})\). To investigate structure of rings, we started with the additive structure. Proved that \((na)\cdot (mb)=(mn) (a\cdot b)\). Recalled the main properties of the order of elements: \(nx=mx\) if and only if \(n\equiv m \pmod{{\rm ord}(x)}\). Defined the characteristic of a ring. Proved that if the lcm of order of elements of \(A\) is a positive integer \(n\), then the characteristic of \(A\) is \(n\). If the lcm of the orders of elements of \(A\) is infinite, then the characteristic of \(A\) is zero. For a unital ring, we proved that the characteristic of \(A\) is the order of \(1_A\) given it is finite. If the order of \(1_A\) is infinite, then \(A\) is of characteristic zero. Mentioned a few examples: \(\mathbb{Z}\) is of characteristic zero. \(\mathbb{Z}_n\) is of characteristic \(n\). Recalled from group theory that \({\rm ord}(x_1,\ldots,x_n)={\rm lcm}({\rm ord}(x_1),\ldots,{\rm ord}(x_n))\). Used this to find the characteristic of \(\mathbb{Z}_9\times \mathbb{Z}_{12}\times \mathbb{Z}_8\). Whenever you are studying a mathematical sturcture, you must ask yourself what are the maps that preserve that structure. In group theory, you had group homomorphisms. In ring theory, you have ring homomorphism. We defined what a ring homomorphism is. Proved that if \(A\) is a unital ring, then \(c:\mathbb{Z}\to A, \quad c(n):=n 1_A\) is a ring homomorphism. Pointed out that a ring homomorphism is also an additive group homomorphism. Recalled what the kernel of a group homomorphism is. Argued why \(\ker c={\rm char}(A) \mathbb{Z}\). Recalled what an equivalence relation is, the congruence modulo \(n\) is an equivalence relation, the equivalence class of \(a\) is \(a+n\mathbb{Z}\), and this is called a coset of the subgroup \(n\mathbb{Z}\). 8/8. (3), Chapter 22 and 23 In the third lecture, we recalled the definition of the ring \(\mathbb{Z}_n\) of integers modulo \(n\), and proved that the map \(c_n : \mathbb{Z} \to \mathbb{Z}_n, \quad c_n(a) = [a]_n\) is a ring homomorphism. Next, we considered two positive integers \(m, n\) and showed that the map \(c_{n,m} : \mathbb{Z}_n \to \mathbb{Z}_m, \quad c_{n,m}([a]_n) = [a]_m\) is a well-defined ring homomorphism if and only if \(m \mid n\). We used this result to prove the Chinese Remainder Theorem (CRT): if \(\gcd(m, n) = 1\), then the map \(\mathbb{Z}_{mn} \to \mathbb{Z}_m \times \mathbb{Z}_n, \quad [a]_{mn} \mapsto ([a]_m, [a]_n)\) is a ring isomorphism. We then recalled the definition of units in a ring and proved that the set of units forms a group under multiplication. Several examples were discussed to illustrate this concept. We defined a division ring and a field, and showed that \(\mathbb{Z}_n^\times = \{ [a]_n \mid \gcd(a, n) = 1 \}.\) This led us to define Euler’s \(\phi\)-function and prove that \(\phi(p^k) = p^{k-1}(p - 1)\) for any prime \(p\) and positive integer \(k\). 8/11. (4), Chapter 23 and 26 In the fourth lecture, we proved that \(\phi(mn) = \phi(m)\phi(n)\) whenever \(\gcd(m, n) = 1\), and discussed how this result allows us to compute Euler’s \(\phi\)-function for any positive integer using its prime factorization. Using the characterization of units in \(\mathbb{Z}_n\), we showed that \(\mathbb{Z}_n\) is a field if and only if \(n\) is prime. We then examined the equation \(2x^2 - x = 0\) and attempted to solve it over \(\mathbb{Q}\), \(\mathbb{Z}\), and \(\mathbb{Z}_{15}\). This example highlighted the importance of avoiding zero-divisors. We defined what an integral domain is and proved that every field is an integral domain, though the converse does not always hold. In particular, we showed that \(\mathbb{Z}_n\) is an integral domain if and only if \(n\) is prime. We proved that a finite integral domain is always a field. Along the way, we showed that an integral domain has the cancellation property. Returning to the equation \(2x^2 - x = 0\) in \(\mathbb{Z}_{15}\), we learned how to apply the Chinese Remainder Theorem to reduce the problem to solving it in \(\mathbb{Z}_3\) and \(\mathbb{Z}_5\), and then lifting the solutions back to \(\mathbb{Z}_{15}\). Finally, we proved that the characteristic of any integral domain is either 0 or a prime number. We concluded the lecture by introducing the idea behind the field of fractions of an integral domain, which we will explore further in the next lecture. 8/13. (5), Chapter 26 Defined the field of fractions of an integral domain, and formulated its universal property. We had our first exam. 8/15. (6), Chapter 26 and 27 Proved the universal property of the field of fractions. Gave an pseudo algorithm to prove that \(Q(D)\simeq F\). Applied this method to show that \(Q(\mathbb{Z}[i])\simeq \mathbb{Q}[i]\). Defined the ring of polynomials over a ring \(A\). Defined the degree and the leading coefficient of a polynomial. Proved that over an integral domain, \(\deg(fg)=\deg f+\deg g\). Mentioned that if there are zero-divisors, then this formula does not hold. Used this to show that if \(D\) is an integral domain, then (1) \(D[x]\) is an integral domain, and (2) \(D[x]^\times=D^\times\). Next, we defined a nilpotent element. Pointed out that if \(a\) is nilpotent and \(A\) is commutative, then every multiple of \(a\) is also nilpotent. Also mentioned that sum of two nilpotent elements in a commutative ring is nilpotent. We did not prove this. Use the binomial expansion to show this. Proved that if \(a\) is nilpotent, then \(1-a\) is a unit. Used this to show that \(1-2x\) is a unit in \(\mathbb{Z}_{16}[x]\) and find its inverse. Mentioned that in general, the group of units of \(A[x]\) consists of polynomials of the form \(a_0+a_1x+\cdots+a_nx^n\) such that \(a_0\in A^\times\) and \(a_1,\ldots,a_n\) are nilpotents. Finally, we formulated the Frobenius homomorphism. It will be proved in the next lecture. 8/18. (7), Chapter 27 and 28 Proved that if \( A \) is a commutative ring of characteristic \( p \), where \( p \) is a prime, then the map \( f: A \to A \), defined by \( f(a) = a^p \), is a ring homomorphism—known as the Frobenius homomorphism. This was used to prove Fermat's Little Theorem. We noted that the polynomials \( x \) and \( x^p \) over \( \mathbb{Z}_p \) define the same function from \( \mathbb{Z}_p \) to itself. Next, we defined the evaluation-at-\( \alpha \) homomorphism: if \( A \subseteq B \) are unital commutative rings and \( \alpha \in B \), then \( \phi_\alpha: A[x] \to B \), given by \( \phi_\alpha(f) := f(\alpha) \), is a ring homomorphism. The kernel of \( \phi_\alpha \) consists of polynomials that vanish at \( \alpha \). We then discussed polynomial long division over arbitrary unital commutative rings, showing that if the leading coefficient of \( g \) is a unit, then there exists a unique pair \( (q, r) \) such that \( f(x) = q(x)g(x) + r(x) \) and \( \deg r < \deg g \). This implies that for any \( a \in A \) and \( f \in A[x] \), there exists a unique \( q \in A[x] \) such that \( f(x) = (x - a)q(x) + f(a) \), leading to a proof of the Factor Theorem. We then proved the Generalized Factor Theorem and noted that it fails for polynomials over \( \mathbb{Z}_{15} \). Finally, we stated that a polynomial of degree \( n \) over an integral domain cannot have more than \( n \) distinct zeros. 8/20. (8), Chapter 28 Proved that if \(D\) is an integral domain, \(f,g\in D[x]\), \(\deg f, \deg g\leq n\), and \(f(a_i)=g(a_i)\) where \(a_i\)'s are \(n+1\) distinct elements of \(D\), then \(f(x)=g(x)\) in \(D[x]\). Using the generalized factor theorem, we proved that \(x^p-x=x(x-1)\cdots (x-(p-1))\) in \(\mathbb{Z}_p[x]\) given \(p\) is prime. Used this to show Wilson's theorem. Defined an irreducible element of a ring. Mentioned that \(n\) is irreducible in \(\mathbb{Z}\) if and only if \(n=\pm p\) where \(p\) is prime (in the classical sense). Discussed irreducible polynomials over a field. Argued why every degree 1 polynomial is irreducible over a field, but \(2x\) is not irreducible over \(\mathbb{Z}\). Used the factor theorem to conclude that a polynomial of degree at least 2 is reducible over a field \(F\) if it has a zero in \(F\). We proved the converse for polynomials of degree 2 or 3 -- also known as the degree 2 or 3 irreducibility criterion. Next, we proved the rational root criterion. Used this to show that a monic integer polynomial has a zero in \(\mathbb{Q}\) if and only if it has a zero in \(\mathbb{Z}\). Deduce that if a monic integer polynomial does not have a zero in \ \(\mathbb{Z}_p\) for some prime \(p\), then it does not have a zero in \(\mathbb{Q}\). Mentioned how we can use Fermat's little theorem to see \(a^{c_kp^k+\cdots+c_0}=a^{c_k+\cdots+c_0}\) for \(a\in \mathbb{Z}_p\) if \(p\) is prime. Using this, one can compute polynomials rather fast over \(\mathbb{Z}_p\). If we get lucky and the polynomial does not have a zero in \(\mathbb{Z}_p\), then we deduce that it does not have a zero in \(\mathbb{Q}\). We mentioned examples like \(x^{51}-x^3+103\) and used \(p=5\). Next, we used the intermediate value theorem to show that a polynomial of odd degree has a zero in the real numbers, and so it is reducible assuming it is not of degree 1. We mentioned the fundamental theorem of algebra (without proof). Finally, we stated the mod-p irredecibility criterion: a monic integer polynomial which is irreducible over \(\mathbb{Z}_p\) is irreducible over \(\mathbb{Q}\), and mentioned our general startegy of proof. 8/22. (9), Chapter 28 Proved three versions of Gauss's Lemma. Established that a primitive polynomial is irreducible over \( \mathbb{Z} \) if and only if it is irreducible over \( \mathbb{Q} \). Demonstrated the mod \( p \) irreducibility criterion and stated Eisenstein's irreducibility criterion. Concluded with some examples. 8/25. (10), Chapter 28 and 30 Proved Eisenstein's criterion and demonstrated that the polynomial \(x^{p-1}+x^{p-2}+\cdots+1\) is irreducible over \(\mathbb{Q}\) when \(p\) is prime. Defined the notaion of an ideal, noting that both \(\{0\}\) and \(A\) are ideals of \(A\) for any ring \(A\). Recalled and proved that every subgroup of \(\mathbb{Z}\) is of the form \(n\mathbb{Z}\) for some non-negative integer \(n\). Established that the set of \(A\)-linear combinations of \(x_1,\ldots,x_n\) form the smallest ideal containing the \(x_i\)'s, assuming \(A\) is a unital commutative ring. Defined principal ideals and Principal Ideal Domains (PIDs), and noted that \(\mathbb{Z}\) is a PID. Showed that the principal ideal generated by \(a\) in a unital commutative ring \(A\) equals \(A\) if and only if \(a\) is a unit. 8/27. (11), Chapter 30 and 31 Proved that a unital commutative ring \(A\) is a field if and only if it has precisely two ideals \(\{0\}\) and \(A\). Proved that the kernel of every ring homomorphism is an ideal. Defined the quotient ring of \(A\) by an ideal \(I\). Proved that \(\pi\colon A\to A/I, \quad \pi(a):=a+I\) is a ring homomorphism and \(\ker \pi=I\). So, a subset of a ring is an ideal if and only if it is the kernel of some ring homomorphism. Looked at the evaluation at \(i\) map from \(\mathbb{Q}[x]\) to the complex numbers \(\mathbb{C}\). Proved that the kernel of this map is the principal ideal generated by \(x^2+1\). Mentioned that this result is part of a more general statement. We concluded the lecture part of the class by stating the fundamental theorem of homomorphisms (also known as the first isomorphism theorem). We had our second exam. 8/29. (12), Chapter 30 and 31 We proved the Fundamental Theorem of Homomorphisms, also known as the First Isomorphism Theorem, and applied it to the evaluation map \( e_{\alpha} : \mathbb{Q}[x] \to \mathbb{C} \). This motivated a deeper study of the ideals of \( F[x] \), where \( F \) is a field. In particular, we established that \( F[x] \) is a Principal Ideal Domain (PID) when \( F \) is a field. Returning to \( \ker e_{\alpha} \), we defined algebraic and transcendental elements, and introduced the minimal polynomial \( m_{\alpha}(x) \) of an algebraic element \( \alpha \). We proved that \( m_{\alpha}(x) \) is irreducible, and that \( f(\alpha) = 0 \) if and only if \( m_{\alpha}(x) \mid f(x) \). Moreover, if \( p \in \mathbb{Q}[x] \) is monic and irreducible with \( p(\alpha) = 0 \), then \( m_{\alpha}(x) = p(x) \). Several examples were presented to illustrate these ideas. We then turned our attention to the image of the evaluation map. By the Fundamental Theorem of Homomorphisms, it suffices to study the quotient ring \( \mathbb{Q}[x]/\langle m_{\alpha} \rangle \). We showed that every element of \( \mathbb{Q}[x]/\langle m(x) \rangle \) has a unique representative of degree less than \( \deg m \), and rephrased this by noting that each element can be written as a \( \mathbb{Q} \)-linear combination of \( 1 + \langle m \rangle, x + \langle m \rangle, \ldots, x^{\deg m - 1} + \langle m \rangle \). This means that \( \mathbb{Q}[x]/\langle m(x) \rangle \) can be viewed as a vector space over \( \mathbb{Q} \) of dimension \( \deg m \). We denoted the image of \( e_{\alpha} \) by \( \mathbb{Q}[\alpha] \), and concluded that \( \dim_{\mathbb{Q}}(\mathbb{Q}[\alpha]) = \deg m_{\alpha} \) when \( \alpha \in \mathbb{C} \) is algebraic. 9/3. (13), Chapter 31 In this lecture, we defined prime ideals and proved that an ideal \( P \triangleleft A \) is prime if and only if the quotient ring \( A/P \) is an integral domain; introduced prime elements and showed that \( p \in A \) is prime if and only if \( \langle p \rangle \) is a prime ideal and \( p \neq 0 \); characterized the prime ideals of \( \mathbb{Z} \), proving that an ideal is prime if and only if it is either \( \{0\} \) or \( p\mathbb{Z} \) for some prime \( p \); noted that \( \{0\} \) is prime precisely when the ring is an integral domain; established the Correspondence Theorem, giving bijections between ideals \( J \) containing \( I \) in \( A \) and ideals of \( A/I \), via \( J \mapsto J/I \) and \( \overline{J} \mapsto \{ x \in A \mid x + I \in \overline{J} \} \); showed that the number of ideals of \( \mathbb{Z}_n \) equals the number of positive divisors of \( n \); defined maximal ideals and proved that \( M \triangleleft A \) is maximal if and only if \( A/M \) is a field; deduced that every maximal ideal is prime; and finally proved that if \( P \triangleleft A \) is prime and \( A/P \) is finite, then \( P \) is maximal. 9/5. (14), Chapter 31 and finite fields First, we proved that for \( a \in D \), where \( D \) is a principal ideal domain (PID), the element \( a \) is irreducible if and only if the ideal \( \langle a \rangle \) is maximal and \( a \neq 0 \). This result was then used to show that in a PID, every irreducible element is prime. Next, we showed that in any integral domain, every prime element is irreducible. Combining these results, we concluded that in a PID, every non-zero prime ideal is maximal. Returning to the evaluation at an algebraic element, we deduced that \( \mathbb{Q}[\alpha] \) is a field whenever \( \alpha \) is algebraic. We then showed that for any irreducible polynomial \( f \in F[x] \), where \( F \) is a field, the quotient ring \( E := F[x]/\langle f \rangle \) is a field. Moreover, the map \( i : F \to E \), defined by \( i(c) := c + \langle f \rangle \), is an injective ring homomorphism, and the image of \( f \) under this map satisfies \( i(f)(\alpha) = 0 \), where \( \alpha := x + \langle f \rangle \). We concluded by constructing a field of order \( 27 \), and outlined a general method for constructing finite fields of order \( p^n \) using irreducible polynomials of degree \( n \) over \( \mathbb{Z}_p \).

My old notes. You are supposed to read these notes and the relevant sections of your book.

Here is the lecture note for the first lecture. In this lecture, we defined ring, and commutative and unital rings. Congruence arithmetic is recalled. The ring of integers modulo \(n\) is introduced. The ring of \(n\)-by-\(n\) matrixes with entries in a ring is introduced. Several examples are mentioned. Here is the lecture note for the second lecture. In this lecture, we showed some basic properties of operations in any ring; defined the characteristic of a unital ring; showed that for any unital ring \(R\) there is a canonical homomorphism \(c:\mathbb{Z}\rightarrow R\); discussed homomorphisms between \(\mathbb{Z}_n\)'s; proved Chinese Remainder Theorem; defined the Euler \(\phi\)-function and proved it is multiplicative; Here is the lecture note for the third lecture. In this lecture, we defined division ring and field; defined integral domain; proved that a finite integral domain is a field; showed that \(\mathbb{Z}_n\) is an integral domain if and only if \(n\) is prime; defined the field of fractions (we will continue its proof in the next lecture). Here is the lecture note for the fourth lecture. In this lecture, we defined the field of fractions of an integral domain, and presented a few examples on this. The ring of polynomials with coefficients in a given ring is also defined. Here is the lecture note for the fifth lecture. In this lecture, we defined degree of a polynomial; showed that \(\deg(fg)=\deg(f)+\deg(g)\) for polynomials over an integral domain; described the group of units of a ring of polynomial. We compared polynomials vs functions; proved Fermat's theorem; and introduced the evaluation homomorphism. Here is the lecture note for the sixth lecture. In this lecture, we studied further the evaluation homomorphisms and their connection with zeros of polynomials; proved the division algorithm; factor theorem; and the fact that a polynomial \(f\) with coefficients in an integral domain has at most \(\deg(f)\) zeros. Here is the lecture note for the seventh lecture. In this lecture, we said what an irreducible element of a unital commutative ring is; described irreducible elements of \(F[x]\) where \(F\) is a field; showed that a degree 2 or 3 polynomial in \(F[x]\) is reducible exactly when it has a zero in \(F\); described a method to effectively find possible rational zeros of a polynomial with integer coefficients; extended the residue maps to the ring of polynomials \(\mathbb{Z}[x]\); showed the following: Suppose \(f(x)\) in \(\mathbb{Z}[x]\) has degree at least 2 and its leading coefficient is 1. If \(c_p(f)\) is irreducible in \(\mathbb{Z}_p[x]\) (for some prime \(p\)), then \(f\) is irreducible in \(\mathbb{Q}[x]\); we showed how Fermat's theorem can be used to find possible zeros of a polynomial with large degrees in \(\mathbb{Z}_p\). Here is the lecture note for the eighth lecture. In this lecture, we proved Gauss's lemma and then used it to prove two irreducibility criteria; one of them is known as Eisenstein Criterion. We also defined what an ideal is. Here is the lecture note for the ninth lecture. In this lecture, we showed that a proper ideal does not have a unit; only ideals of a field \(F\) are 0 and \(F\); ideals of \(\mathbb{Z}\) are of the form \(n\mathbb{Z}\) for some intgere \(n\); defined what a Principal Ideal Domain is; proved that the ring of polynomials \(F[x]\) with coefficients in a field \(F\) is a PID; proved that the kernel of a ring homomorphism is an ideal; defined the quotient ring and the natural quotient map; deduced that a subset of a ring \(R\) is an ideal exactly when it is kernel of a ring homomorphism of \(R\). Here is the lecture note for the tenth lecture. In this lecture, we proved the Fundamental Homomorphism Theorem (also known as the First Isomorphism Theorem); proved \(\mathbb{Z}/n\mathbb{Z}\) is isomorphic to \(\mathbb{Z}_n\) as a ring; studied the evaluation map at an algebraic number via an example: showed that the kernel of \(\phi_{\sqrt{2}}\) is generated by \(x^2-2\); the image of \(\phi_{\sqrt{2}}=\mathbb{Q}[\sqrt{2}]\); and \(\mathbb{Q}[x]/\langle x^2-2\rangle\simeq \mathbb{Q}[\sqrt{2}]\); proved that if \(\alpha\in \mathbb{C}\) is an algebraic number, then there is an irreducible polynomial \(m_{\alpha}(x)\in \mathbb{Q}[x]\) which generates the kernel of the evaluation map \(\phi_{\alpha}:\mathbb{Q}[x]\rightarrow \mathbb{C}\) at \(\alpha\). Here is the lecture note for the eleventh lecture. In this lecture, we proved that the image of the evaluation map \(\phi_{\alpha}\) at an algebraic number \(\alpha\) which is a zero of a polynomial of degree \(d_0\) is \(\mathbb{Q}[\alpha]:=\{c_0+c_1 \alpha+\cdots+ c_{d_0-1} \alpha^{d_0-1}|\hspace{1mm} c_i\in \mathbb{Q}\}\); To prove \(\mathbb{Q}[\alpha]\) is a field, we investigated the ideals \(I\) of a unital commutative ring \(R\) with the property that \(R/I\) is a field; and we proved that \(R/I\) is a field if and only if \(I\) is a maximal ideal of \(R\) (here \(R\) is a unital commutative ring.). Here is the lecture note for the twelfth lecture. In this lecture, we proved if \(D\) is a PID and \(a\in D\setminus \{0\}\), then \(\langle a\rangle\) is maximal if and only if \(a\) is irreducible in \(D\). Using this result we finished the proof of the following theorem. Suppose \(\alpha\in \mathbb{C}\) is an algebraic number. Then there is a polynomial \(m_{\alpha}(x)\in \mathbb{Q}[x]\) such that (1) \(m_{\alpha}(x)\) is irreducible in \(\mathbb{Q}[x]\), (2) for \(f(x)\in \mathbb{Q}[x]\), if \(\alpha\) is a zero of \(f(x)\), then \(m_{\alpha}(x)\) divides \(f(x)\); that means \(f(x)=m_{\alpha}(x)q(x)\) for some \(q(x)\in \mathbb{Q}[x]\). And \(\mathbb{Q}[\alpha]:=\{c_0+c_1\alpha+\cdots+c_{d-1}\alpha^{d-1}|c_i\in \mathbb{Q}\}\) is a field where \(d=\deg m_{\alpha}\). Next we defined what a prime ideal is; and showed that in a unital commutative ring \(R\) an ideal \(I\) is prime if and only if the quotient ring \(R/I\) is an integral domain. In the rest of the lecture we proved the following theorem. Let \(F\) be a field and \(p(x)\) be an irreducible polynomial in \(F[x]\). Then there are a field \(E\), an embedding \(i:F\rightarrow E\), and \(\alpha\in E\) such that \(\alpha\) is a zero of \(i(p)(x)\) (which essentially means \(p(x)\), but viewed as an element of \(E[x]\) via the embedding \(i\).) Here is the lecture note for the thirteenth lecture. In this lecture, we defined what a Unique Factorization Domain is, and proved that a PID is a UFD; the existence part is only proved for \(F[x]\), but you can find the argument for the general case in the lecture note. Along the way certain properties of irreducible elements of a PID is also proved. Here is the lecture note for the fourteenth lecture. In this lecture, we proved that, if \(f(x)\in \mathbb{Z}_p[x]\) is an irreducible polynomial of degree \(d_0\), then there are a field extension \(E\) of \(\mathbb{Z}_p\) and \(\alpha\in E\) such that \(\alpha\) is a zero of \(f(x)\); moreover \(\mathbb{Z}_p[\alpha]=\{c_0+c_1\alpha+\cdots+c_{d_0-1}\alpha^{d_0-1}|\hspace{1mm} c_i\in \mathbb{Z}_p\}\) is a field and \(|\mathbb{Z}_p[\alpha]|=p^{d_0}\). Hence if there is an irreducible polynomial of degree \(d_0\) in \(\mathbb{Z}_p[x]\), then there is a finite field of order \(p^{d_0}\). In the rest of the lecture, we went through some of the old topics.

Homework Homework are due on Thursdays at 9:00 pm. You should post them in GradeScope. Late Homework are not accepted. Selected problems on the each assignment will be graded. Style: A messy and disorganized homework might get no points. You can scan or simply take a clear photo of your homework and upload it. You must select pages corresponding to your solutions of problems during the upload process. If you have not selected pages when the grader begins grading, the grader will not grade your assignment and you will receive a grade of 0 on it. No appeals of this policy will be considered. As a math major, sooner or later you have to learn how to use LaTex. I really encourage you to use Latex to type your solutions. You can use Overleaf. Overleaf is an easy to use and an excellent online LaTeX editor. Reading the sections of the textbook corresponding to the assigned homework exercises is considered part of the homework assignment. You are responsible for material in the assigned reading whether or not it is discussed in the lecture. There will be 5 problem sets. Your cumulative homework grade will be based on the best 4 of the 5. You can work on the problems with your classmates, but you have to write down your own version. Copying from other's solutions is not accepted and is considered cheating. A good portion of the exams will be based on the weekly problem sets. So it is extremely important for you to make sure that you understand each one of them.

Grade Your weighted score is the best of Homework 20%+ exam I 20%+ exam II 20%+ Final 40% Homework 20%+ The best of exams I and II 20%+ Final 60% Your letter grade is determined by your weighted score using the best of the following methods: A+ A A- B+ B B- C+ C C- D F 97 93 90 87 83 80 77 73 70 67 67>  Based on a curve where the median corresponds to the cut-off B-/C+.

Further information There is no make-up exam. No notes, textbooks, calculators and electronic devices are allowed during exams.

Exams. The first exam: Time: Wednesday, August 13, during the lecture. Location: this is an in-class exam. (the second half of the lecture). Topics: All the topics that are discussed in class and Sections 22 and 23 from the main book of the course. Questions are fairly similar to the homework assignments and the examples discussed in the class. Make sure that you know how to solve anyone of them. Practice: In addition to reviewing your homework assignments, in-class examples, and the relevant textbook chapters, you may find the following practice exams helpful for further preparation: Summer 2017, PDF . Winter 2019, PDF . Exam: The second exam: Time: Wednesday, August 27, during the lecture. Location: this is an in-class exam. (the second half of the lecture). Topics: All the topics that are discussed in class and Sections 22, 23, 26, 27, 28, and 30. Make sure that you know all the topics covered in the lecture notes; for instance the fact that \(x^p=x\) for any \(x\in \mathbb{Z}_p\) if \(p\) is prime, and \(F[x]\) is a PID when \(F\) is a field are not covered in the mentioned Sections, but they might be part of the exam. But the Fundamental Homomorphism Theorem (which is discussed in Section 30) is not part of this exam. Any topic discussed in the mentioned Sections except the Fundamental Homomorphism Theorem might be part of the exam. Questions are fairly similar to the homework assignments and the examples discussed in the class. Make sure that you know how to solve anyone of them. Practice: In addition to reviewing your homework assignments, in-class examples, and the relevant textbook chapters, you may find the following practice exams helpful for further preparation: Summer 2017, PDF . Winter 2019, PDF . Exam: The final exam: Date: Saturday, September 6, 8:00-10:59 Location: the usual classroom. Topics: All the topics that were discussed in class, your homework assignments, and relevant examples and exercises in your book. Questions are fairly similar to the homework assignments and the examples discussed in the class. Make sure that you know how to solve anyone of them. Summary of lectures You may find this summary useful -- you can skip the part on Unique Factorization Domains. Practice: In addition to reviewing your homework assignments, in-class examples, and the relevant textbook chapters, you may find the following practice exams helpful for further preparation: Summer 2017, PDF . Winter 2019, PDF .