Abstract Algebra 2 Quiz

The smallest field containing an integral domain R. It's formed by considering all possible fractions with numerators and denominators from R, excluding zero as a denominator.

The unique smallest subfield of a field F. It's the subfield contained within every other subfield of F.

An ideal M in a commutative ring with unity R is maximal if there is no other ideal I that strictly contains M and is also strictly contained in R.

A polynomial f(x) is reducible over a field F if it can be factored into two non-constant polynomials with smaller degrees. Otherwise, it's irreducible.

A polynomial f(x) is irreducible over a field F if its only factors are a non-zero constant and a polynomial of the same degree.

A proper ideal P in a commutative ring with unity is prime if the product of two elements being in P implies at least one of the elements is already in P.

Every maximal ideal in a commutative ring with unity is also a prime ideal.

The field of rational functions is formed by considering all possible fractions where the numerator and denominator are polynomials with coefficients from a base field.

A polynomial is reducible if it can be factored into two polynomials of lower degree. A polynomial is irreducible if it cannot be factored into two polynomials of lower degree. These are important concepts for understanding the structure of polynomials and their roots.

A polynomial is said to be irreducible over a field F if it cannot be factored into two polynomials of lower degree with coefficients in F. This means the polynomial cannot be broken down into simpler expressions using only the elements of F.

A polynomial of degree 1 is always irreducible over any field. Here's why: A linear polynomial (ax + b) can only be factored by pulling out a constant factor, which doesn't reduce the degree.

A polynomial of degree 2 or 3 is irreducible over a field if it has no roots in that field. This means you can't find a value within the field that makes the polynomial equal to zero. Since a polynomial of degree n has at most n roots, if it has no roots, it cannot be factored into linear polynomials.

If a polynomial in Z[x] (integers) has a rational root r/s (in lowest form), then r must divide (be a factor of) the constant term of the polynomial and s must divide (be a factor of) the leading coefficient.

The content of a polynomial is the greatest common divisor (GCD) of its coefficients. A polynomial is called primitive if its content is a unit (1 or -1) in the ring of coefficients.

Gauss's Lemma states that the product of two primitive polynomials is also primitive. This is a key result that shows that the concept of primitivity is preserved under multiplication.

Eisenstein's Criterion is a powerful tool for proving the irreducibility of certain polynomials over the rationals. It states that if a polynomial has a prime p that divides all coefficients except the leading coefficient and does not divide the constant term, then it's irreducible over the rationals.