Polynomals 1 More propertes of polynomals Recall that, for R a commutatve rng wth unty (as wth all rngs n ths course unless otherwse noted), we defne R[x] to be the set of expressons n =0 a x, where a R, wth the understandng that two such expressons agree f they dffer by terms of the form 0x k. Alternatvely, we could dentfy a polynomal wth an nfnte sequence a 0, a 1,..., such that a R and only fntely many of the a are non-zero. Addton and multplcaton of polynomals are defned as follows: a x + b x Y = (a + b )x ; ( ) ( ) a x b x = a b j. k +j=k Note that, wth ths defnton, x x j = x +j, and hence x = } x x {{ x}. tmes Thus the two meanngs of x are consstent. We wll use symbols such as f, g, p, q for polynomals, unlke the more usual notatons f(x), etc. n order to emphasze that polynomals are formal or symbolc objects. We wll dscuss the varous ways n whch we can thnk of polynomals as functons later. It s routne to check that (R[x], +) s an abelan group. To check that t s a rng, we must check that multplcaton s assocatve and commutatve, that the left dstrbutve law holds (we don t have to check both laws snce multplcaton s commutatve), and that there s a unty. We wll just check assocatvty. Wth f = a x ; g = b x ; ; h = c x, 1
a calculaton shows that and smlarly (fg)h = l f(gh) = l +j+k=l +j+k=l (a b j )c k x l, a (b j c k ) x l. Thus (fg)h = f(gh) snce multplcaton n R s assocatve. Note that R s a subrng of R[x], wth ( ) r a x = ra x, and n partcular 1 R s the unty n R[x]. Remark 1.1. The defnton of addton and multplcaton n R[x] s essentally forced by requrng assocatvty, commutatvty, and dstrbutvty. For example, we must have ax + bx = (a + b)x. Lkewse, we must have (ax )(bx ) = abx +j, provded that we nterpret x as (x), the product of the rng element x wth tself tmes. As prevously stated, the degree of a polynomal f = a x s the largest nteger d such that a d 0. The degree of the zero polynomal 0 s undefned. Note that, f a R, a 0, then deg a = 0, and n fact the subrng R of R[x] s gven by R = {f R[x] : deg f = 0 or f = 0}. We shall sometmes refer to the elements of R as constant polynomals or constants. If f = d =0 a x s a polynomal of degree d and g = e =0 b x s a polynomal of degree e and, say, d > e, then clearly the degree of f + g s d. But f f and g both have the same degree d, the term (a d + b d )x d mght be 0, f b d = a d, and hence n ths case deg(f + g) < d, or s undefned f g = f. Thus, f f, g, f + g 0, then deg(f + g) max(deg f, deg g). Smlarly, f f and g are as above, then the hghest degree term of fg s a d b e x d+e, unless a d b e = 0. Hence, f f, g, fg 0, then deg(fg) deg f + deg g. 2
Example 1.2. In (Z/6Z)[x], (2x + 1)(3x 2 + 1) = 3x 2 + 2x + 1, snce the leadng term (2x)(3x 2 ) = 0, and hence the product does not have the expected degree 1 + 2 = 3. Even worse, 2(3x 2 + 3) = 0. In (Z/4Z)[x], (2x+1) 2 = 4x 2 +4x+1 = 1. Thus, not only does (2x+1) 2 not have the expected degree, but we also see that 2x + 1 s a unt,.e. there are rngs R such that the group of unts (R[x]) s larger than the unts R n R. Polynomals n several varables can be defned smlarly. For example, an element of R[x 1, x 2 ],.e. a polynomal n the two varables x 1, x 2, s an expresson of the form,j 0 a jx 1 xj 2, where the a j R, and only fntely many are nonzero. By groupng such terms n powers of x 2, we see that R[x 1, x 2 ] = R[x 1 ][x 2 ]. In other words, a polynomal n x 1 and x 2 s the same thng as a polynomal n x 2 whose coeffcents are polynomals n x 1. Smlarly R[x 1, x 2 ] = R[x 2 ][x 1 ] by groupng n powers of x 1. Inductvely, we can defne polynomals n n varables va R[x 1,..., x n ] = R[x 1,..., x n 1 ][x n ]. 2 Polynomals as functons A polynomal wth real coeffcents f = a x defnes a functon f : R R by defnng f(t) = a t. (Typcally, we speak of x as the varable, not just some formal symbol.) We can do the same thng n a general rng: gven r R, we defne the evaluaton ev r of a polynomal f = a x at r, and wrte t as ev r (f) or sometmes as f(r), by the formula ev r (f) = a r R. Informally, ev r (f) s obtaned from f by pluggng n r for x. In ths way, an element f R[x] also defnes a functon from R to R, whch we denote by E(f), va the formula E(f)(r) = f(r) = ev r (f). For example, f a R R[x] s a constant polynomal, then ev r (a) = a and E(f) s the constant functon from R to tself whose value s always a. Lkewse, ev r (x) = r and E(x): R R s the dentty functon. To te ths n wth rng theory, we have 3
Proposton 2.1. () For all r R, the functon ev r : R[x] R s a homomorphsm. () The functon E s a rng homomorphsm from R[x] to R R, the rng of all functons from R to tself (wth the operatons of pontwse addton and multplcaton). Proof. () We must check that, for all f, g R[x], ev r (f + g) = ev r (f) + ev r (g); ev r (fg) = ev r (f) ev r (g). Wth f = a x and g = b x, ev r (f) + ev r (g) = a r + b r = (a + b )r = ev r (f + g). Here of course we can add as many terms of the form 0x k as are needed to make sure that the sums for f and g have the same lmts. For multplcaton, wth f and g as above, ( ) ( ) ev r (f) ev r (g) = a r b r = a b j r +j =,j = a b j r k = ev r (fg). k +j=k Fnally, ev r (1) = 1, so ev r takes the unty n R[x] to the unty n R. () We must check that, for all f, g R[x], E(f + g) = E(f) + E(g); E(fg) = E(f)E(g). To check for example that the functons E(f + g) and E(f) + E(g)are equal, we must check that they have the same value at every r,.e. that E(f + g)(r) = (E(f) + E(g))(r) = E(f)(r) + E(g)(r) for every r R, where the second equalty s just the defnton of pontwse addton of functons. By defnton, E(f + g)(r) = ev r (f + g) = ev r (f) + ev r (g), by Part (), and so E(f + g)(r) = ev r (f) + ev r (g) = E(f)(r) + E(g)(r) as clamed. Fnally we must check that E(1) = 1, where the rght hand 1 s the unty n R R. Here E(1)(r) = ev r (1) = 1 for all r, and hence E(1) s the constant functon f : R R whose value at every r R s 1. Ths s the unty n R R. 4
In more down to earth terms, Part () above just says that every polynomal n R[x] defnes a functon from R to R, and that the operatons of polynomal addton and multplcaton correspond to pontwse addton and multplcaton respectvely (and that the constant polynomal 1 corresponds to the constant functon 1). One reason (among many) that we want to be somewhat pedantc about ths setup s the followng observaton: For R = R, the homomorphsm E : R[x] R R s njectve: ths just says that a polynomal functon determnes the polynomal tself (.e. ts coeffcents) unquely. We wll gve an algebrac argument for ths fact, n much more generalty, soon. Of course, the homomorphsm E : R[x] R R s defntely not surjectve, snce most functons from R to R are not polynomals. There are varous generalzatons of the homomorphsm ev r : 1. In the case of the polynomal rng R[x 1,..., x n ] n n varables, gven r 1,..., r n R, we can evaluate f R[x 1,..., x n ] at (r 1,..., r n ). Ths gves a homomorphsm ev r1,...,r n : R[x 1,..., x n ] R, as well as a homomorphsm E : R[x 1,..., x n ] R Rn. In other words, a polynomal n n varables defnes a functon of n varables,,.e. a functon R n R. Note that ev r1,...,r n can be defned nductvely: vewng R[x 1,..., x n ] as R[x 1,..., x n 1 ][x n ] and r n R R[x 1,..., x n 1 ], ev rn s a homomorphsm ev rn : R[x 1,..., x n 1 ][x n ] R[x 1,..., x n 1 ], and by repeatng ths constructon successvely we get ev r1,...,r n = ev r1 ev rn : R[x 1,..., x n ] R. 2. Suppose that R s a subrng of a rng S and that s S. Then we can restrct ev s to the subrng R[x] of S[x] to defne a homomorphsm ev s : R[x] S. For example, we mght want to evaluate a polynomal wth real coeffcents on a complex number such as. As we have seen, the mage of ev s s a subrng of S, and s denoted R[s]. By defnton, snce ev s (a) = a for all a R and ev s (x) = s, the subrng R[s] of S contans R and s. In fact, { } R[s] = a s : a R. Clearly, every subrng of S contanng R and s contans s for all nonnegatve ntegers, hence contans a s for all a R and thus 5
contans R[s]. Thus: R[s] s the smallest subrng of S contanng R and s. For example, the rngs Z[], Z[ 3 2] are of ths type. Of course, snce 2 = 1, gven a n Z, we can rewrte n a n n as a sum only nvolvng actual ntegers (n even) as well as ntegers tmes (n odd), so every expresson of the form n a n n s actually of the form a + b where a, b Z. A smlar remark holds for Z[ 3 2], usng the fact that ( 3 2) n s always of the form a, b 3 2, or c( 3 2) 2 for ntegers a, b, c dependng on whether n s congruent to 0, 1, or 2 mod 3. More generally, gven s 1,..., s n S, we can defne ev s1,...,s n : R[x 1,..., x n ] S. The mage of ev s1,...,s n s a subrng of S, denoted by R[s 1,..., s n ], and t s the smallest subrng of S contanng R and s 1,..., s n. 3. Suppose that ϕ: R S s a homomorphsm. Then we can defne a homomorphsm from R[x] to S[x], whch for smplcty we also denote by ϕ, by applyng ϕ to all of the coeffcents of f. Explctly, f f = a x, we set ϕ(f) = ϕ(a )x. It s easy to check from the defnton of polynomal multplcaton and the fact that ϕ preserves addton and multplcaton that ϕ: R[x] S[x] s also a rng homomorphsm. We have tactly used one example of ths already: f R s a subrng of S, then R[x] s a subrng of S[x]. For another mportant example, let π : Z Z/nZ be the projecton of an nteger to ts congruence class mod n. Then we get a homomorphsm π : Z[x] (Z/nZ)[x], whch conssts n reducng the coeffcents of an nteger polynomal mod n. 4. We can also amalgamate the examples above: gven a ϕ: R S and an element s S, we can defne ev ϕ,s = ev s ϕ. In other words, gven the polynomal f R[x], frst apply the homomorphsm ϕ to the coeffcents of f to vew t as a polynomal n S[x], then evaluate t at s. One general theme of ths course s as follows: let F be a feld (typcally F s Q, R, C, F p ) and let f F [x]. Then we want to fnd a root or zero of f (sometmes we say we want to solve the equaton f = 0). Ths means we want to fnd an element r F such that ev r (f) = f(r) = 0. By experence, 6
such as wth the polynomal x 2 + 1 R[x] or x 2 2 Q[x], sometmes we cannot fnd such an r wthn F. In ths case, we look for a larger feld E,..e a feld contanng F as a subfeld, and an element s E such that ev s (f) = 0. 7