Abstract Algebra 2 Quiz

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Questions and Answers

What is the smallest field that contains an integral domain called?

  • Field of Unity
  • Prime Field
  • Field of Quotients (correct)
  • Integral Field

If char R = 0, which of the following structures exists within R?

  • A field isomorphic to Z
  • A maximal ideal
  • A prime ideal
  • A subring isomorphic to Q (correct)

Which of the following is true regarding maximal and prime ideals?

  • Maximal and prime ideals can be equal.
  • Every maximal ideal is prime. (correct)
  • Every prime ideal is maximal.
  • Prime ideals can never be maximal.

Which statement describes a polynomial f that is irreducible in F[x]?

<p>It cannot be factored nontrivially into polynomials of smaller degrees. (C)</p> Signup and view all the answers

If char F = p > 0, which of the following subfields can F contain?

<p>Subfield isomorphic to Zp (C)</p> Signup and view all the answers

A polynomial that is reducible in F[x] must:

<p>Have factors that can be constant or of smaller degrees. (A)</p> Signup and view all the answers

What is a prime ideal P in a ring R characterized by?

<p>Whenever ab ∈ P, then either a ∈ P or b ∈ P. (B)</p> Signup and view all the answers

What is true about the field of rational functions Frac(F[x])?

<p>It includes rational expressions formed from polynomials. (B)</p> Signup and view all the answers

Which of the following statements is true regarding a polynomial of degree 1?

<p>It is irreducible. (A)</p> Signup and view all the answers

If a polynomial f has degree greater than or equal to 2 and has a root c, what can be concluded about its factors?

<p>f has a factor of the form x - c. (C)</p> Signup and view all the answers

What does Gauss’s lemma state concerning primitive polynomials?

<p>The product of primitive polynomials is primitive. (A)</p> Signup and view all the answers

When is a polynomial irreducible in Q[x]?

<p>If it is a primitive polynomial. (C)</p> Signup and view all the answers

Eisenstein’s Criterion provides a necessary condition for what type of polynomial?

<p>Irreducible over Q. (A)</p> Signup and view all the answers

Which of the following is a polynomial that is irreducible over Q?

<p>x^3 - 2 (B)</p> Signup and view all the answers

What is true about a polynomial that is reducible in Z[x]?

<p>It might become irreducible in some other polynomial rings. (C)</p> Signup and view all the answers

According to the Mod p Test, what can be inferred if a polynomial reduces to an irreducible polynomial in Zp [x]?

<p>It is irreducible in Q[x]. (D)</p> Signup and view all the answers

Flashcards

Field of Quotients (or Field of Fractions)

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.

Prime Subfield

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

Maximal Ideal

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.

Reducible Polynomial

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.

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Irreducible Polynomial

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.

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Prime Ideal

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.

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Maximal Ideal Implies Prime Ideal

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

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Field of Rational Functions

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.

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Reducible vs. Irreducible Polynomials

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.

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Irreducible over a field F

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.

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Polynomials of Degree 1 are Irreducible

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.

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Irreducibility of Degree 2 & 3 Polynomials

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.

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Rational Roots Theorem

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.

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Content of a Polynomial

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.

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Gauss's Lemma

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.

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Eisenstein's Criterion

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.

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Study Notes

Abstract Algebra 2

  • Every integral domain can be embedded in a field.
  • A field of quotients is the smallest field containing an integral domain.
  • The field of quotients of a ring R is denoted as Frac(R) = { a/b | a, b ∈ R, b ≠ 0}.
  • Q = Frac(Z)
  • Frac(F[x]) = F(x)

Fields and Subfields

  • Every field has a unique, smallest subfield, called its prime subfield.
  • If the characteristic of a ring R is 0, then R has a subring isomorphic to Z.
  • If the characteristic of a ring R is a positive integer n, then R has a subring isomorphic to Zn.
  • If the characteristic of a field F is 0, then F has a subfield isomorphic to Q.
  • If the characteristic of a field F is a prime p, then F has a subfield isomorphic to Zp

Maximal Ideals

  • Let R be a commutative ring with unity.
  • An ideal M of R is maximal if for any ideal I in R, M ⊆ I ⊆ R implies M = I or M = R.
  • If M is a maximal ideal of R, then R/M is a field.

Prime Ideals

  • Let P be a proper ideal of R. P is prime if for all a, b ∈ R, if ab ∈ P, then a ∈ P or b ∈ P.
  • In any commutative ring with a unity, every maximal ideal is prime.

Reducible and Irreducible Polynomials

  • A polynomial f is reducible in F[x] (over a field F) if it can be factored into polynomials of smaller degrees.
  • A polynomial f in F[x] (over a field F) is irreducible if it only factors into a nonzero constant and a polynomial of the same degree.
  • Rules for determining irreducibility:
    • If the degree of a polynomial is 1, then it is irreducible.
    • If the degree of a polynomial is greater than 2, has a root c in F, then ( x − c ) is a factor.
    • If the degree of a polynomial is 2 or 3 and has no roots in F, then it is irreducible.
  • f∈Z[x] is irreducible over Q (in lowest form): r is a root, then r is a divisor of the constant term, s is a divisor of the leading coefficient.
  • The content of a polynomial, C(f), is the greatest common divisor of its coefficients.
  • f is a primitive polynomial if C(f) is a unit.
  • Gauss's Lemma: the product of primitive polynomials is primitive.
  • If f is irreducible in Z[x], then f is irreducible in Q[x]. The converse is true only if f is primitive.
  • Mod p test: If f reduces to an irreducible polynomial in Zp[x], then f is irreducible in Q[x].
  • Eisenstein's Criterion: a specific condition to determine irreducibility over Q, using prime numbers.

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