Abstract Algebra 2 PDF

Abstract Algebra 2 Date: Every integral domain can be embedded in a field. We can find the smallest field that contains an integral domain. This is called its field of quotients or field of fractions. Let F...

Abstract Algebra 2 Date: Every integral domain can be embedded in a field. We can find the smallest field that contains an integral domain. This is called its field of quotients or field of fractions. Let F = Frac(R) be the field of quotients of R. Then F = { ab≠1 | a, b œ R, b ”= 0 }. Q = Frac(Z) and Y - Z - ]f (x) -- ^ Frac(F [x]) = F (x) = f, g œ F [x], g = ” 0 [ g(x) -- \ called the field of rational functions. Every field F has a unique, smallest subfield that is a subfield contained in every subfield of F. This is called its prime subfield. Let R be a ring with unity. If char R = 0, then R has a subring isomorphic to Z. If char R = n > 0, then R has a subring isomorphic to Zn. Let F be a field with unity. If char F = 0, then F has a subfield isomorphic to Q. If char F = p > 0, then F has a subfield isomorphic to Zp. Let R be a commutative ring with unity. Let M be a proper ideal of R. M is maximal … for an ideal I in R, M ™ I ™ R ∆ M = I or M = R. M is maximal … R M is a field. Let P be a proper ideal of R. P is prime … for a, b œ R, a œ P or b œ P whenever ab œ P. P is prime … R P is an integral domain in R, every maximal ideal is prime. A polynomial f is reducible in F [x] (or over F ) if it factors nontrivial into polynomials of smaller degrees. f œ F [x] is irreducible in F [x] (or over F ) if only factors into a nonzero constant and a polynomial of same degree. f is reducible over F f is irreducible over F nonzero constant f = g·h f = c·g deg f > deg g, deg h deg f = deg g © 2024, Kenneth James Nuguid, DMS, PUP Manila 3 Abstract Algebra 2 Date: If f is a nonzero constant (unit) then f is reducible (not irreducible). If deg f = 1, then f is irreducible. If deg f Ø 2 and f has a root c then f has a factor x ≠ c If deg f = 2 or deg f = 3 and f has no roots in F , then f is irreducible. If f factors into polynomials of smaller degree, then f is reducible. If f factors only as a product of a nonzero constant and a polynomial of same degree, then f is irreducible. If f œ Z[x], then rs œ Q (in lowest form) is a root, then r is a divisor of the constant term while s is a divisor of the leading coefficient. C(f ) or content of f is the gcd of its coefficients. f is said to be primitive iff C(f ) is a unit in R Gauss’s lemma states that the product of primitive polynomials is primitive. To obtain a polynomial in Z[x] we can multiply f by the lcd of the coefficients. f is irreducible in Z[x] ∆ f is irreducible in Q[x]. The converse is true only if f is primitve. 2x + 2 = 2(x + 1) which is not primitive is irreducible in Q[x] but not in Z[x] Let p be a prime. Mod p Test: If f reduces to an irreducible polynomial in Zp [x] then f is irreducible in Q[x]. Converse is not necessarily true. f = x3 ≠ x + 1 is irreducible over Z2. Thus, f = x3 ≠ 2x2 + x + 1 reduces to fp = x3 + x + 1 in Z2 [x]. Then f is irreducible over Q. In fact, any polynomial in Z[x] that reduces to x3 + x + 1 is irreducible over Q. Eisenstein’s Criterion: Let f = an xn + · · · + a1 x + a0 œ Z[x]. ÷ p (prime) s.t. p | a0 ,... , an≠1 , p - an , p2 - a0 ∆ f is irreducible over Q. There is always a polynomial of degree n which is irreducible over Q, which is xn ≠ p where p is a prime. © 2024, Kenneth James Nuguid, DMS, PUP Manila 4