Schaum's Outline of Discrete Mathematics, Third Edition (Schaum's Outlines)

APP. B] ALGEBRAIC SYSTEMS 445 Note first thatJis a subring ofR. Also,Jis a subgroup (necessarily normal) of the additive group o ...

446 ALGEBRAIC SYSTEMS [APP. B EXAMPLE B.15 (a) The ringZof integers is the classical example of a unique factorization domain. T ...

APP. B] ALGEBRAIC SYSTEMS 447 Theorem B.12: LetKbe an integral domain. ThenK[t]under the operations of addition and multiplicati ...

448 ALGEBRAIC SYSTEMS [APP. B Corollary B.16 also tells us thatf(a)=0 if and only if the remainderr=r(t)≡0. Accordingly: Corolla ...

APP. B] ALGEBRAIC SYSTEMS 449 Theorem B.21: Letfandgbe polynomials inK[t], not both zero. Then there exists a unique monic polyn ...

450 ALGEBRAIC SYSTEMS [APP. B EXAMPLE B.17 Letf(t)=t^4 − 3 t^3 + 6 t^2 + 25 t−39. Find all the roots off(t)given thatt= 2 + 3 ii ...

APP. B] ALGEBRAIC SYSTEMS 451 B.3.LetS=N×N. Let∗be the operation onSdefined by (a,b)∗(a′,b′)=(aa′,bb′). (a) Show that∗is associa ...

452 ALGEBRAIC SYSTEMS [APP. B (1) Proof thatis well-defined: We have([a])=f(a). Sincea∈S, we havef(a)∈f(S). Hence([a])∈f(S), ...

APP. B] ALGEBRAIC SYSTEMS 453 (c) We have 2^2 =4, 2^3 =8, 2^4 =1. Hence| 2 |=4 and gp(2)={1, 2, 4, 8}. Also, 72 =4, 7^3 = 4 ∗ 7 ...

454 ALGEBRAIC SYSTEMS [APP. B B.9.Letσandτbe the following elements of the symmetric groupS 6 : σ= ( 123456 315462 ) and τ= ( 12 ...

APP. B] ALGEBRAIC SYSTEMS 455 Fig. B-8 (a) AsymmetryσofSis a rigid one-to-one correspondence betweenSand itself. (Here rigid mea ...

456 ALGEBRAIC SYSTEMS [APP. B (c) Sincee(a)=a, we havee∈Ha. Supposeg, g′∈Ha. Then(gg′)(a)=g(g′(a))=g(a)=a; hencegg′∈Ha. Also,g−^ ...

APP. B] ALGEBRAIC SYSTEMS 457 B.18.SupposeF:G→G′is a group homomorphism. Prove: (a)f(e)=e′;(b)(fa−^6 )=f(a)−^1. (a) Sincee=eeand ...

458 ALGEBRAIC SYSTEMS [APP. B (c) Substitute each of the ten elements ofZ 10 intof(x)to see which elements yield 0. We have: f( ...

APP. B] ALGEBRAIC SYSTEMS 459 POLYNOMIALS OVER A FIELD B.26.Supposef(t)= 2 t^3 − 3 t^2 − 6 t−2. Find all the roots off(t)knowing ...

460 ALGEBRAIC SYSTEMS [APP. B B.31.Prove Theorem B.18: Supposef(t)is a polynomial over a fieldK, and deg(f)=n. Thenf(t)has at mo ...

APP. B] ALGEBRAIC SYSTEMS 461 Supposepdividesfgbut notf. Sincepis irreducible, the polynomialsfandpmust then be relatively prime ...

462 ALGEBRAIC SYSTEMS [APP. B B.41. LetAbe a nonempty set with the operation∗defined bya∗b=a, and assumeAhas more than one eleme ...

APP. B] ALGEBRAIC SYSTEMS 463 B.53. Consider the symmetric groupS 4. Letα= ( 1234 3421 ) andβ= ( 1234 2431 ) . (a) Findαβ,βα,α^2 ...

464 ALGEBRAIC SYSTEMS [APP. B RINGS B.69. Consider the ringZ 12 ={ 0 , 1 ,..., 11 }of integers modulo 12. (a) Find the units ofZ ...