Modern Algebra: Polynomials and Irreducibility

Questions and Answers

Which of the following statements is true regarding algebraic numbers?

Algebraic numbers can be expressed using arithmetic and radicals. (correct)

Algebraic numbers cannot be the roots of any polynomial.

Transcendental numbers are algebraic.

All complex numbers are algebraic.

Which of the following is classified as a transcendental number?

3/4

π (correct)

√-1

The square root of 2

What does the set A of all algebraic numbers form?

A field (correct)

A vector space

A group under addition

A ring with no unity

In the context of algebraic numbers, what does it mean for a number to be algebraic over Q?

It is the root of a polynomial with rational coefficients. Signup and view all the answers

What is the significance of the unsolved question about algebraic numbers until the early 1800s?

Whether all algebraic numbers can be expressed using radicals. Signup and view all the answers

What is the dimension of the vector space Q(√2, i) over Q?

4 Signup and view all the answers

Which of the following correctly identifies the minimal polynomial of ζ = e^(2πi/3) over Q?

x^2 + x + 1 Signup and view all the answers

What does the notation [Q(√3) : Q] represent?

The vector space dimension of Q(√3) over Q Signup and view all the answers

What is the degree of the extension Q(√2) over Q?

2 Signup and view all the answers

If r is algebraic and not in field F, what is true about its minimal polynomial over F?

It is unique up to scalar multiplication. Signup and view all the answers

What represents the degree of a polynomial function f(x) = an x^n + an−1 x^(n−1) + ... + a0?

The highest non-zero power n Signup and view all the answers

In the context of the polynomial set F[x], what is the implication if the coefficients are from a field F?

The set behaves as a vector space Signup and view all the answers

Why are some numbers considered 'uncomputable'?

They cannot be expressed using arithmetic and radicals Signup and view all the answers

Which of the following statements about the polynomial f(x) = 5x^4 - 18x^2 - 27 is correct?

It can have integer coefficients only when expressed in Z[x] Signup and view all the answers

How can the roots of low-degree polynomials typically be represented?

By a combination of arithmetic and radicals Signup and view all the answers

What is the primary limitation of polynomial expressions regarding uncomputable numbers?

They do not allow representation of certain real numbers Signup and view all the answers

Which of the following sets includes polynomials with integer coefficients?

Z[x] Signup and view all the answers

If a polynomial cannot be expressed using arithmetic and radicals, what alternative method might be necessitated?

Introducing special symbols like π or e Signup and view all the answers

Why is the field of rational numbers not algebraically closed?

It contains a polynomial with roots not in the set. Signup and view all the answers

What is a characteristic feature of complex roots of polynomials with real coefficients?

They appear in pairs as complex conjugates. Signup and view all the answers

What does the fundamental theorem of algebra state about the field of complex numbers?

Every polynomial can be completely factored into linear factors in this field. Signup and view all the answers

Which of the following polynomials does NOT split into linear factors over the rational numbers?

$f(x) = x^2 + 1$ Signup and view all the answers

How can we define an algebraically closed field?

A field where every non-constant polynomial has a root in that field. Signup and view all the answers

What can be concluded if a field does not allow every polynomial to split into linear factors?

The field is not algebraically closed. Signup and view all the answers

If a polynomial has real coefficients, which of the following is necessarily true about its complex roots?

They are conjugates of each other. Signup and view all the answers

Why is the field of real numbers not algebraically closed?

It contains polynomials without real roots. Signup and view all the answers

Which condition is necessary for a polynomial of degree greater than 1 to be classified as reducible over a field F?

It has a root in F. Signup and view all the answers

What does Eisenstein's criterion state about a polynomial's irreducibility?

It implies that if a polynomial is irreducible over Z, it is also irreducible over Q. Signup and view all the answers

For the polynomial $x^2 + x + 1$, what can be concluded regarding its irreducibility?

It is irreducible over Q since Eisenstein's criterion fails. Signup and view all the answers

Which of the following polynomials is irreducible over Q according to the given content?

$x^3 - 2$ Signup and view all the answers

In the context of vector spaces, what is a necessary characteristic for a set to qualify as a vector space over Q?

Closure under vector addition. Signup and view all the answers

Which of the following statements about the polynomial $x^4 + 5x^2 + 4$ is correct?

It can be factored into polynomials of lower degree. Signup and view all the answers

What is the relationship between a polynomial's degree and its reducibility over a field?

Degree 2 or 3 polynomials are reducible if they have roots in the field. Signup and view all the answers

Which of the following is true about the field $Q(√2)$ as a vector space over Q?

It has dimension two and a basis of {1, √2}. Signup and view all the answers

Study Notes

Polynomials

A polynomial is a function expressed as ( f(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_2 x^2 + a_1 x + a_0 ), where ( n ) is the degree of ( f ) and ( a_i ) are coefficients from a field ( F ).
The symbol ( F[x] ) represents the set of polynomials over the field ( F ), while ( Z[x] ) denotes polynomials with integer coefficients.

Radicals

Roots of low-degree polynomials can be expressed with arithmetic and radicals.
Some numbers cannot be represented by radicals, leading to the concept of "uncomputable" numbers, as many complex numbers cannot be expressed algorithmically.

Algebraic Numbers

A complex number is algebraic over ( Q ) if it is the root of a polynomial in ( Z[x] ).
The set of all algebraic numbers forms a field, while numbers like ( \pi ), ( e ), and the golden ratio ( \phi ) are transcendental.
Numbers expressible using natural numbers, arithmetic, and radicals are algebraic.

Hasse Diagrams

Hasse diagrams illustrate relationships between number sets: ( N ) (natural numbers), ( Q ) (rationals), ( R ) (reals), ( A ) (algebraic numbers), and ( C ) (complex numbers).

Complex Numbers

A field ( F ) is algebraically closed if every polynomial ( f(x) \in F[x] ) has all its roots in ( F ).
( C ) is algebraically closed, meaning polynomials in ( Z[x] ) completely factor over ( C ).

Complex Conjugates

Complex roots appear in conjugate pairs; if ( r = a + bi ) is a root, then ( \bar{r} = a - bi ) is also a root.

Irreducibility

A polynomial ( f(x) ) is reducible over ( F ) if it can be factored into lower-degree polynomials ( g(x)h(x) ).
Example of reducible polynomials: ( x^2 - x - 6 = (x + 2)(x - 3) ) and ( x^4 + 5x^2 + 4 = (x^2 + 1)(x^2 + 4) ).
Non-examples: ( x^3 - 2 ) is irreducible over ( Q ) due to lack of roots in ( Q ).

Eisenstein’s Criterion for Irreducibility

Stein's criterion provides a method to determine irreducibility for polynomials in ( Z[x] ) with specific conditions related to a prime ( p ).
If ( f(x) = a_n x^n + \ldots + a_0 ), ( f ) is irreducible if ( p ) divides ( a_i ) for ( i < n ) and does not divide ( a_n ) or ( a_0 ).

Extension Fields as Vector Spaces

An extension field ( Q(r) ) can be viewed as a vector space over ( Q ).
Example: ( Q(\sqrt{2}) ) is a 2-dimensional vector space over ( Q ).
The degree of an extension, denoted ( [E : F] ), is the dimension of ( E ) as a vector space over ( F ).

Minimal Polynomials

The minimal polynomial of an algebraic element ( r ) over ( F ) is the irreducible polynomial for which ( r ) is a root.
Examples:
- The minimal polynomial of ( \sqrt{2} ) over ( Q ) is ( x^2 - 2 ).
- The minimal polynomial of ( i ) over ( Q ) is ( x^2 + 1 ).

Degree Theorem

The degree of the extension ( Q(r) ) corresponds to the degree of the minimal polynomial of ( r ) over ( Q .

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Description

This quiz focuses on the concepts of polynomials and their irreducibility as discussed in Lecture 6.3 of Math 4120, Modern Algebra. Students will explore definitions, properties, and examples related to polynomials in this advanced mathematical context.