Fields - fields of fractions of integral domains, characteristics of a field, algebraic extensions, splitting fields, simple extensions - One Line Questions
1.
If a field F has characteristic p > 0, then for any elements a, b in F, which property holds? —
(a + b)^p = a^p + b^p
2.
Let K be an extension of F. If α ∈ K is algebraic over F, then the degree of the minimal polynomial of α over F is equal to: —
[F(α):F]
3.
What is the characteristic of the field Zp, where p is a prime number? —
p
4.
What is the characteristic of the field of rational numbers (Q)? —
0
5.
Consider the extension Q(√2) over Q. What is the degree of this extension? —
2
6.
What is the characteristic of a field F if 1 + 1 = 0 in F? —
2
7.
Consider the field extension Q(i) over Q, where i^2 = -1. What is the degree [Q(i):Q]? —
2
8.
Consider the extension Q(√2, √3) over Q. What is [Q(√2, √3):Q]? —
4
9.
Let K be the splitting field of x^3 - 2 over Q. What is [K:Q]? —
6
10.
Consider the extension Q(∛2, ω) over Q, where ω is a primitive cube root of unity. What is the degree of this extension? —
6
11.
What is a field extension? —
A field K such that F is a subfield of K.
12.
The set of all algebraic numbers over Q forms: —
A field, but not algebraically closed.
13.
Let α be an element of an extension field K of F. If α is algebraic over F, then F(α) is: —
A finite extension of F.
14.
What is the splitting field of the polynomial x^p - a over F, where F has characteristic p > 0? —
A purely inseparable extension.
15.
Let K be an extension of F. If K = F(α) for some element α in K, what is K called? —
A simple extension of F
16.
If every element of an extension field K of F is algebraic over F, then K is called: —
An algebraic extension of F
17.
Let K be an extension field of F. If [K:F] is finite, then K is: —
An algebraic extension of F.
18.
The splitting field of an irreducible polynomial p(x) of degree n over a field F is: —
An extension of degree n over F if p(x) is separable.
19.
If F is a field, what is its characteristic? —
Either 0 or a prime number.
20.
What is the relationship between algebraic extensions and finite extensions? —
Every finite extension is algebraic, but the converse is not always true.
21.
The field of fractions of an integral domain R is constructed by: —
Forming equivalence classes of fractions a/b where a, b are in R and b ≠ 0.
22.
If α is transcendental over F, what is the degree of the extension F(α) over F? —
Infinite
23.
What is the defining property of an integral domain? —
It is a commutative ring with unity and no zero divisors.
24.
If F is a field of characteristic p > 0, what is true about the additive subgroup generated by the multiplicative identity 1? —
It is isomorphic to Zp.
25.
What is the characteristic of the field Z_4 (integers modulo 4)? —
It is not a field.
26.
If F is a field, what can be said about the characteristic of any subfield of F? —
It is the same as the characteristic of F.
27.
If a field has characteristic p > 0, what can be said about the polynomial x^p - x? —
Its roots form a finite field.
28.
If [K:F] = 1, what is the relationship between K and F? —
K = F
29.
Let K be a finite extension of F, and let L be a finite extension of K. Then: —
L is a finite extension of F, and [L:F] = [L:K][K:F].
30.
If K is a splitting field of a polynomial p(x) over F, and L is an intermediate field such that F ⊆ L ⊆ K, is L necessarily the splitting field of p(x) over F? —
No, not necessarily.
31.
Let F = Z_p(t) be the field of rational functions in t over Z_p. What is the characteristic of F? —
p
32.
Let K be the splitting field of x^2 - 2 over Q. What is K? —
Q(√2)
33.
What is the field of fractions of the integral domain Z[√2] = {a + b√2 | a, b ∈ Z}? —
Q(√2)
34.
Let K be the splitting field of x^2 + 1 over R. What is K? —
The complex numbers C.
35.
If K is the splitting field of a separable polynomial p(x) over F, then [K:F] is equal to: —
The degree of p(x).
36.
The degree of a field extension K over F, denoted [K:F], is defined as: —
The dimension of K as a vector space over F.
37.
Let F be a field. The field of fractions of F[x] is isomorphic to: —
The field of rational functions F(x).
38.
Let F be a field. What is the field of fractions of the integral domain F[x]? —
The field of rational functions in x over F, denoted F(x).
39.
What is the minimal polynomial of an element α algebraic over a field F? —
The monic polynomial of least degree in F[x] that has α as a root.
40.
What is the field of fractions of the ring of integers Z? —
The rational numbers Q.
41.
Which of the following is NOT an integral domain? —
The ring of 2x2 matrices with real entries.
42.
Which of the following is a field of fractions of the integral domain of integers (Z)? —
The set of rational numbers (Q)
43.
What is a splitting field of a polynomial p(x) over a field F? —
The smallest extension field K of F such that p(x) splits into linear factors in K[x].
44.
What is the smallest field containing F and all the roots of a polynomial p(x) ∈ F[x]? —
The splitting field of p(x) over F.
45.
An element α in an extension field K of F is said to be algebraic over F if: —
There exists a non-zero polynomial p(x) with coefficients in F such that p(α) = 0.
46.
If K is an algebraic extension of F, and L is an algebraic extension of K, then L is algebraic over F. This is known as the: —
Transitivity of algebraic extensions.
47.
Let α be algebraic over F. The degree of F(α) over F is equal to the degree of the minimal polynomial of α over F. This statement is: —
True.
48.
Let F be a field of characteristic 0. If α is algebraic over F, then the minimal polynomial of α over F is: —
Unique and irreducible.
49.
Consider the extension Q(∛2) over Q. Is this a finite extension? —
Yes, because the minimal polynomial of ∛2 over Q is x^3 - 2, which has degree 3.
50.
For any non-constant polynomial p(x) in F[x], does a splitting field always exist? —
Yes, it always exists.