Fields

Question 1

Subfield

Answer 1

Let F and K be fields and suppose that F is a subset of K and that addition and multiplication on F are the same as those on K. Then we say that F is a subfield of K.

Question 2

Maximal ideal

Answer 2

Let R be a ring and let I be an ideal of R. We say that I is the maximal ideal if I=/R and for any ideal J or R where I is a subset of J is a subset of R, then either I=J or J=R.

Question 3

Maximal ideal in Z

Answer 3

Let n be a natural number with n not equal to 1. The is the maximal ideal of Z iff n is prime

Question 4

Maximal ideal in F

Answer 4

Let F be a field and let m(X) be an element of F[x] be a non constant polynomial. Then is the maximal ideal of F[X] iff m(x) is an irreducible polynomial.

Question 5

The quotient ring of a commutative ring by a maximal ideal is a field

Answer 5

Let R be a commutative ring with one and let I be an ideal of R. Then R/I is a field iff I is a maximal.

Question 6

Characteristic

Answer 6

Let F be a field. The characteristic of F is the smallest mEN s.t. m1=0 in F, if such m exists, and is equal to zero if m1=/0

Question 7

Characteristic proposition

Answer 7

Let F be field. The charF=0 or charF=p, where p is prime.

Question 8

Prime subfield

Answer 8

Consider a subring S of F consisting of the elements of the form m1, where m E Z.
Let E= {m1/n1 : m,n E Z, n1=/0}, so that E= {r/s : r, s E S, s=/0}. Then E is a subring as it satisfies the subring test and it also has a multiplicative inverse, therefore E is the prime subfield of F}

Question 9

Field extension

Answer 9

Let F and K be fields where F is a subset of K. We say that K is extension field of F, or that F C_ K is a field extension

Question 10

Finite extension

Answer 10

Let FC_K be a field extension. We say that FC_K is a finite extension if K is finite dimensional as a vector space over F.

Question 11

F[a]

Answer 11

Let F C_ K be a field extension and let aEK. We define F[a} to be the subring of K consisting of all elements of the the form f(a), where f(x) E F[X]

Question 12

F(a)

Answer 12

F(a) to be the subfield of K consisting of all elements of the form f(a)/g(a) where f(x), g(x) E F[x] and g(a)=/0

Question 13

Simple field extension

Answer 13

We refer to F C_ F(a) and we say F(a) is obtained by adjoining a to F.

Question 14

Algebraic

Answer 14

Let F C_ K be field extension and let aeK. We say that a is algebraic over F if there exists a non-zero polynomial f(x) e F[x] such that f(a)=0

Question 15

Minimal polynomial

Answer 15

Let F C_ K be a field extension and let aeK be algebraic over F. The monic polynomial ma(x) e F[X] s.t. {f(x) e F[x}; f(a)=0} = is the minimal polynomial of a over F.