Modern Algebra 2 Chapter 4 vocab (copy)

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Field

A set F with two binary operators, multiplication and addition

- F is an abelian group under addition (closure, associativity, identity)

- F is an abelian group under multiplication

- F obeys the distributive laws

Examples of fields

Rational, real, and complex numbers and prime integers

polynomial

A polynomial in the indeterminate x on a field F is an expression of the form a_nx^n + .. + a_1x + a_0, where n>= 0 is an integer and a_i is a field element for all i. a_i are coefficients.

Leading coefficient

if a_n != 0, then a_n is the leading coefficient

Monic polynomial

a polynomial with a leading coefficient of 1

Constant polynomial

Polynomial with a degree of 0 (a_i = 0 for all i > 0)

F[x]

the set of all polynomials over F in the indeterminate x

Equality

two polynomials p(x) = a_nx^n + .... + ax + a_0, q(x) = b_nx^n + ... +bx + b_0 are equal if a_i = b_i, for all i

Division Algorithm for integers

If a,b are integers, and b != 0, then there exists integers q and r such that a = bq + r, where 0 <= r < | b |

Division Algorithm for polynomials

If a(x), b(x) are in F[x], and b(x) != 0 then there exist unique polynomials q(x), r(x) in F[x] such that a(x) = b(x)q(x) + r(x), where r(x) = 0 or deg(r(x)) < deg(b(x))

f(α)

the evaluation of f(x) at alpha (α). f(α) = a_nα^n + ... + a_1α + a_0

Remainder Theorem

If f(x) is in F[x], then there exists q(x) is in F[x] such that

f(x) = (x - α)q(x) + f(α) (that is, the remainder on dividing f(x) by x - α is f(x)).

Combination theorem for polynomials

Let f(x), g(x), h(x) are in F[x]. If h(x) | f(x) and h(x) | g(x), then

h(x) | a(x)f(x) + b(x)g(x), for all a(x)b(x) in F[x]

Relatively prime

f(x), g(x) in F[x] are relatively prime if their GCD = 1

Euclid's Lemma

if f(x) | g(x)h(x) and (f(x), g(x)) = 1, then f(x) | h(x)

Irreducible polynomial

p(x) is irreducible iff whenever p(x) = f(x)g(x), f(x), g(x) in F[x] one of f(x) or g(x) is a constant polynomial.

Reducible polynomial

p(x) is reducible iff p(x) = a(x)b(x) with 1 <= deg a(x) < deg p(x) and 1<= deg b(x) < deg p(x)

Unique factorizatioin theorem

Any nonconstant polynomial with coefficients in the field F can be expressed as an element of F times a product of monic polynomials, each of which is irreducible over the field F. This expression is unique except for the order in which the factors occur

Root of multiplicity

Number of times a root appears in a polynomial. Let f(x) be in F[x]. An element c in F is said to be a root of multiplicity m>=1 of f(x) if (x-c)^m | f(x) but (x-c)^m+1 \ f(x)

Extension field

Let E and F be fields. If F is a subset of E and has the operations of addition and multiplication induced by E, E is the extension field of F

Subfield

Let E and F be fields. If F is a subset of E and has the operations of addition and multiplication induced by E, F is the subfield of E.

Congruence class

The set {b(x) in F[x] | a(x) congruent mod p(x)} is the congruence class of a(x) [a(x)]

Field Isomorphism

Let F_1 and F_2 be fields. A function phi: F_1 -> F_2 is called an isomorphism of fields if it is one-to-one and onto such that

phi(a + b) = phi(a) + phi(b)

phi(ab) = phi(a) * phi(b)

for all a, b in F_1

Primitive polynomial

A polynomial with integer coefficients (Z[x]) if 1 and -1 are the only common divisors of its coefficients

Gauss's Lemma

Let f(x) be in Z[x]. If f(x) = a_1(x)b_1(x) for some a_1(x), b_1(x) in Q[x], then f(x)=a(x)b(x) for some a(x), b(x) in Z[x] and a_1(x) = alpha a(x), b_1(x) = beta b(x) for some alpha, beta in Q

Rational Roots Theorem

Let f(x) = a_n*x^n + .. + a_1x + a_0 in Z[x]. If alpha = r/s in Q is a rational root of f(x) with (r, s) = 1, then r | a_0 and s | a_n.

Eisenstein's Criterion

Let f(x) = a_nx^n + a_(n-1)x^(n-1) + ... + a_1x + a_0 be a polynomial with integer coefficients. If there is a prime p such that p | a_0, p | a_1, ... p | a_(n-1) but p \ a_n and p^2 \ a_0 then f(x) is irreducible over Q.

Fundamental theorem of Algebra

If f(x) in C[x] is a nonconstant polynomial then f(x) has a root in C. In particular, no polynomial in C[x] of degree greater than 1 is irreducible over C.