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1 2. R I N G S A N D P O LY N O M I A L S The study of vector spaces and linear maps between them naturally leads us to the study of rings, in particular the ring of polynomials F[x] and the ring of (n n)-matrices M n (F). 2.1 Rings. Definition 2.1: A non-empty set R with two binary operations + and is a ring if (R, +, 0) is an abelian groups, the multiplication is associative and the distribution laws hold: for all a, b, c R, (a + b)c = ac + bc and a(b + c) = ab + ac. The ring R is called commutative if for all a, b R we have ab = ba. Examples 2.2: (a) Any field is a commutative ring. (b) Z, Z[i], F[x] are commutative rings. (c) M n (F), the set of (n n)-matrices over F, and Hom(V, V), the set of linear self-maps for any vector space V, are non-commutative rings when n > 1 or dim(v) > 1. (d) For A M n (F) the set of polynomials F[A] evaluated on A form a commutative subring of M n (F). Definition 2.3: A map φ : R S between two rings is a ring homomorphism if for all r, r R: φ(r + r ) = φ(r) + φ(r ) and φ(rr ) = φ(r)φ(r ). A bijective ring homomorphism is a ring isomorphism. Example 2.4: We can reinterpret (part of) Theorem 1.14 to say that T B [T] B defines an isomorphism of rings from Hom(V, V) to M n (F) where n is the dimension of V. Definition 2.5: A non-empty subset I of a ring R is an ideal if for all s, t I and r R we have s t T and sr, rs I. Warning: Some books insist on rings having a multiplicative identity 1 and on ring homomorphisms taking 1 to 1. If we do not insist on 1

2 2 rings having units, then any ideal is a subring. (Note that in a unital ring, any ideal that contains 1 is the whole ring.) Examples 2.6: (a) mz is an ideal in Z. Indeed, every ideal in Z is of this form! [To prove this, let m be the smallest non-zero integer in the ideal I and prove that I = mz.] (b) The set of diagonal matrices in M n (R) is closed under addition and multiplication (i.e. it is a subring) but is not an ideal. Ideals are to rings what normal subgroups are to groups in the sense that the set of coset R/I inherit a ring structure from R if I is an ideal. For r, r R define (r + I) + (r + I) := (r + r ) + I and (r + I)(r + I) := rr + I. We leave it as an exercise to check that these operations are welldefined. First Isomorphism Theorem 2.7: The kernel ker(φ) := φ 1 (0) of a ring homomorphism φ : R S is an ideal, its image im(φ) is a subring of S, and φ induces an isomorphisms of rings R/ker(φ) im(φ). Proof: Exercise.[Show that the underlying isomorphisms of abelian groups is compatible with the multiplication, i.e. is a ring homomorphism.] 2.2 Polynomial rings. We will discuss polynomials over a field F in more detail. The first goal is to show that there is a division algorithm for polynomial rings. With hte help of this we will able to show the important property that every ideal in F[x] is generated by one element q(x). Theorem 2.8: Division algorithm for polynomials Let f (x), g(x) F[x] be two polynomials with g(x) = 0. Then there exists q(x), r(x) F[x] such that f (x) = q(x)g(x) + r(x) and deg g(x) > deg r(x). Proof: If deg f (x) < deg g(x), put q(x) = 0, r(x) = f (x). Assume now that deg f (x) deg g(x) and let f (x) = a n x n + a n 1 x n a 0 g(x) = b k x k + b k 1 x k b 0

3 3 Then ( deg f (x) a ) n x n k g(x) < n b k By induction on deg f deg g, there exist s(x), t(x) such that f (x) a n b k x n k g(x) = s(x)g(x) + t(x) and deg g(x) > deg t(x). Hence put q(x) = a n b k x n k + s(x) and r(x) = t(x). Corollary 2.9: For all f (x) F[x] and a F, f (a) = 0 (x a) f (x). Proof: By Theorem 2.8 there exist q(x), r(x) such that f (x) = q(x)(x a) + r(x) where r(x) is constant (as deg r(x) = 0). Evaluating at a gives and hence r = 0. f (a) = 0 = q(a)(a a) + r = r Corollary 2.10: If deg f n then f has at most n roots. Proof: This follows from Corollary 2.9 and induction. Let a(x), b(x) F[x] be two polynomials. Let c(x) be a monic polynomial of highest degree dividing both a(x) and b(x) and write c = (a, b). Proposition 2.11: Let a, b F[x] be non-zero polynomials and let (a, b) = c. Then there exist s, t F[x] such that: a(x)s(x) + b(x)t(x) = c(x). Proof: If c = 1, divide a and b by c. We may thus assume deg(a) deg(b) and (a, b) = 1, and will proceed by induction on deg(a) + deg(b). By the Division Algorithm there exist q, r F[x] such that a = qb + r with deg(b) > deg(r). Then deg(a) + deg(b) > deg(b) + deg(r) and (b, r) = 1. If r = 0 then b(x) = λ is constant since (a, b) = 1. Hence a(x) + b(x)(1/λ)(1 a(x)) = 1. Assume r = 0. Then by the induction hypothesis, there exist s, t F[x] such that bs + rt = 1.

4 4 Hence, bs + (a qb)t = 1 and at + b(s qt ) = 1 So, we may put t = t and s = s qt. Exercise: Prove that every ideal I F[x] is generated by one element. In other words, there exists a polynomial c(x) such that I = { f (x)c(x) f (x) F[x]}. 2.3 Evaluating polynomials on matrices. We will now turn to evaluating polynomials at (n n)-matrices. Given a matrix we can associate two special polynomials to it, its charcateristic and its minimal polynomial. As we will see these will encode much of the information of interest. Let A M n (F) and f (x) = a k x k + + a 0 F[x]. Then f (A) := a k A k + + a 0 I M n (F). Since A p A q = A q A p and λa = Aλ for all f (x), g(x) F[x], we have that f (A)g(A) = g(a) f (A); Av = λv f (A)v = f (λ)v. Lemma 2.12: For all A M n (F), there exists a polynomial f (x) F[x] such that f (A) = 0. Proof: Note that the dimension dim M n (F) = n n < is finite. Hence {I, A, A 2,, A k } as a subset of M n (F) is linearly dependent for k n 2. Hence there exist scalars a i F such that a k A k + + a 0 I = 0, and f (x) = a k x k + + a 0 is an annihilating polynomial. Remark: We can express much of the above in terms of ring theory as follows. For any (n n)-matrix A, the assignment f (x) f (A) defines a ring homomorphism E A : F[x] M n (F). As F[x] is commutative, so is the image of E A and in particular f (A)g(A) = g(a) f (A) for all polynomials f and g. Our next step is to determine the unique monique polynomial generating the kernel of E A.

5 5 2.4 Minimal and characteristic polynomials. Definition 2.13: The minimal polynomial of A, denoted by m A (x), is the monic polynomial p(x) of least degree such that p(a) = 0. Theorem 2.14: If f (A) = 0 then m A f. Furthermore, m A is unique (hence showing that m A is well-defined). Proof: By the division algorithm, Theorem 2.8, there exist polynomials q, r with deg r < deg m A such that f = qm A + r. Evaluating both sides at A gives r(a) = 0. By the minimality property of m A, r 0 and m A divides f. To show uniqueness, let m be another monic polynomial of minimal degree and m(a) = 0. Then by the above m A m. Also m and m A must have the same degree, and so m = am A for some a F. By monicness of both polynomials it follows that a = 1 and m = m A. Definition 2.15 The characteristic polynomial of A is defined as χ A (x) = det(a xi). Lemma 2.16: χ A (x) = ( 1) n x n + ( 1) n 1 tr(a)x n det A. Proof: a 11 x a 12 a 1n. χ A (x) = det a.. 21 a2n..... a n1 a n2 a nn x By induction, the coefficient of x n is ( 1) n and the coefficient of x n 1 is ( 1) n 1 tr A. The constant term is χ A (0) = det(a). Definition 2.17: λ is an eigenvalue of A if there exists v = 0 F n such that Av = λv. Theorem 2.18: λ is an eigenvalue of A λ is a root of χ A (x) λ is a root of m A (x)

6 6 Proof: χ A (λ) = 0 det(a λi) = 0 A λi is singular v = 0 : (A λi)v = 0 v = 0 : Av = λi m A (λ)v = m A (A)v = 0 m A (λ) = 0 (as v = 0) m A (x) = (x λ)g(x) Vice versa, assume λ is a root of m A. Then m A (x) = g(x)(x λ) for some polynomial g. By minimality of m A, we have g(a) = 0. Hence there exists w F n such that g(a)w = 0. Put v = g(a)w then and v is a λ-eigenvector for A. (A λi)v = m A (A)v = 0, One of our next goals is to prove that χ A annihilates A and hence that m A χ A. We finish this section by recording how to translate back what we have learnt about matrices to the world of linear maps. In particular we will show that it makes sense to speak of a minimal and characteristic polynomial of a linear transformation. Let C, P, A be (n n)-matrices such that C = P 1 AP. Then m C (x) = m A (x) for: f (C) = f (P 1 AP) = P 1 f (A)P = 0 for all polynomials f. thus and similarly m C (A) = m C (C) = 0 and m A (C) = m A (A) = 0 so that m C m A and m A m C and therefore m A = m C as both are monic. Definition/Lemma 2.19: Let V be a finite dimensional vector space and T : V V a linear transformation. Define the minimal polynomial of T as m T (x) = m A (x) where A = B [T] B with respect to some basis B of V. As m A (x) = m P 1 AP(x) the definition of m T (x) is independent of the choice of basis. Definition/Lemma 2.20: For a linear transformation T : V V define its characteristic polynomial as χ T (x) = χ A (x)

7 7 where A = B [T] B with respect to some basis B of V. As χ A (x) = χ P 1 AP(x) the definition of χ T (x) is independent of the choice of basis. A. Appendix: Algebraically closed fields. It will be conveninet and illuminating to be able to refer to algebraically closed fields. Definition 2.21: A field F is algebraically closed if every non-constant polynomial in F[x] has a root in F. Theorem 2.22: Fundamental Theorem of Algebra C is algebraically closed. [We will not be able to show this in this course. However, you should be able to prove it using complex analysis by the end of this term. Hint: consider g(z) = 1/ f (z). If f (z) has no roots in C, it is holomorphic and bounded on C, which leads to a contradiction.] Definition 2.23: An algebraically closed field F containing F with the property that there does not exist a smaller algebraically closed field L with F L F is called the algebraic closure of F. Theorem 2.24: Every field F has an algebraic closure F. [The proof is beyond this course but it will be convenient to appeal to this result.] Challenge: Prove that no finite field is algebraically closed. [Hint: imitate the standard proof that there are infinitely many primes.]

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