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]?

It cannot be factored nontrivially into polynomials of smaller degrees. Signup and view all the answers

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

Subfield isomorphic to Zp Signup and view all the answers

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

Have factors that can be constant or of smaller degrees. Signup and view all the answers

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

Whenever ab ∈ P, then either a ∈ P or b ∈ P. Signup and view all the answers

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

It includes rational expressions formed from polynomials. Signup and view all the answers

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

It is irreducible. 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?

f has a factor of the form x - c. Signup and view all the answers

What does Gauss’s lemma state concerning primitive polynomials?

The product of primitive polynomials is primitive. Signup and view all the answers

When is a polynomial irreducible in Q[x]?

If it is a primitive polynomial. Signup and view all the answers

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

Irreducible over Q. Signup and view all the answers

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

x^3 - 2 Signup and view all the answers

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

It might become irreducible in some other polynomial rings. 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]?

It is irreducible in Q[x]. Signup and view all the answers

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 Z_n.
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 Z_p

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 Z_p[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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Description

Test your knowledge on concepts from Abstract Algebra 2, including integral domains, fields, and ideals. This quiz covers embedding integral domains into fields, subfields, maximal ideals, and prime ideals. Challenge yourself on the intricate relationships within algebraic structures.