Fields
We will cover following topics
Fields
A ring \(F\) whose non-zero elements form an abelian multiplicative group is called a field.
For example, the rings \(Q\), \(R\) and \(C\) are fields.
Subfield: Any subset \(F'\) of a field \(F\), which is itself a field with respect to the field structure of \(F\), is called a subfield of \(F\).
For example, \(Q\) is a subfield of the fields \(R\) and \(C\); also, \(R\) is a subfield of \(C\).
Prime Field: A field \(F\) which has no proper subfield F0 is called a prime field.
Field Extensions
A field extension is a pair of fields \(E \subseteq F\) such that the operations of \(E\) are those of \(F\) restricted to \(E\). In this case, \(F\) is an extension field of \(E\) and \(E\) is a subfield of \(F\).
For example, under the usual notions of addition and multiplication, the complex numbers are an extension field of the real numbers; the real numbers are a subfield of the complex numbers.
Quotient Fields
The field of fractions of an integral domain is the smallest field in which it can be embedded.
The elements of the field of fractions of the integral domain \(R\) are equivalence classes written as \(\dfrac{a}{b}\) with
\(a\) and \(b\) in \(R\) and \(b \neq 0\)
PYQs
Fields
1) Let \(a\) be an irreducible element of the Euclidean ring \(R\), then prove that \(R/(a)\) is a field.
[10M]
2) Let \(F\) be a field and \(F[x]\) denote the ring of polynomial over \(F\) in a single variable \(X\). For \(f(X), g(X) \in F[X]\) with \(g(X) \neq 0\), show that there exist \(q(X), r(X) \in F[X]\) such that degree \(r(X)\)< degree \(g(X)\) and \(f(X)=q(X) \cdot g(X)+r(X)\).
[2017, 20M]
3) Let \(K\) be a field and \(K[X]\) be the ring of polynomials over \(K\) in a single \(X\) for a polynomial \(f \in K[X]\). Let \((f)\) denote the ideal in \(K[X]\) generated by \(f\). Show that \((f)\) is a maximal ideal in \(K[X]\) if and only iff is an irreducible polynomial over \(K\).
[2016, 15M]
4) Show that every algebraically closed field is infinite.
[2016, 10M]
5) Let \(K\) be an extension of a field \(F\). Prove that the element of \(K\) which are algebraic over \(F\) form a subfield of \(K\). Further if \(F \subset K \subset L\) Fare fields \(L\) is algebraic over \(K\) and \(K\) is algebraic over \(F\), then prove that \(L\) is algebraic over \(F\).
[2016, 20M]
6) Do the following sets form integral domains with respect to ordinary addition and multiplication? Is so, state if they are fields:
i) The set of numbers of the form \(b \sqrt{2}\) with \(b\) rational
(ii) The set of even integers
(iii) The set of positive integers
[2015, 5+6+4=15M]
7) Show that \(Z_{7}\) is a field. Then find \(([5]+[6])^{-1}\) and \((-[4])^{-1}\) in \(Z_{7}\).
[2014, 15M]
8) Show that the set \(\left\{a+b \omega : \omega^{3}=1\right\}\), where \(a\) and \(b\) are real numbers, is a field with respect to usual addition and multiplication.
[2014, 15M]
9) Show that the set of matrices \(S=\left\{\left( \begin{array}{cc}{a} & {-b} \\ {b} & {a}\end{array}\right) a, b \in R\right\}\) is a field under the usual binary operations of matrix addition and matrix multiplication. What are the additive and multiplicative identities and what is the inverse of \(\left( \begin{array}{cc}{1} & {-1} \\ {1} & {1}\end{array}\right)\)? Consider the map \(f : C \rightarrow S\) defined by \(f(a+i b)=\left( \begin{array}{cc}{a} & {-b} \\ {b} & {a}\end{array}\right).\) Show that \(f\) is an isomorphism. (Here \(R\) is the set of real numbers and \(C\) is the set of complex numbers).
[2013, 10M]
10) Consider the polynomial ring \(Q[x]\). Show \(p(x)=x^{3}-2\) is irreducible over \(Q\). Let \(I\) be the ideal \(Q[x]\) in generated by \(p(x)\). Then show that \(\dfrac{Q[x]}{I}\) is a field and that each element of it is of the form \(a_{0}+a_{1} t+a_{2} t^{2}\) with \(\alpha_{0}, a_{1}, a_{2}\) in \(Q\) and \(t=x+I\).
[2010, 15M]
11) Prove that every Integral Domain can be embedded in a field.
[2008, 15M]
12) Show that any maximal ideal in the commutative ring \(F \left[ x \right]\) of polynomials over a field \(F\) is the principal ideal generated by an irreducible polynomial.
[2008, 15M]
13) \(1+\sqrt{-3}\) in \(Z \left[ \sqrt{-3} \right]\) is an irreducible element, but not prime.
Justify your answer.
[2007, 15M]
14) The residue class ring \(\dfrac{Z}{(m)}\) is a field iff \(m\) is a prime integer.
[2004, 15M]
15) Show that \(Q(\sqrt{3}, i)\) is a splitting field for \(x^{5}-3 x^{3}+x^{2}-3\) is the field of rational numbers.
[2003, 15M]
16) Prove that \(x^{2}+x+4\) is irreducible over \(F\) the field of integers modulo 11 and prove further that \(\dfrac{F[x]}{\left(x^{2}+x+4\right)}\) is a field having 121 elements.
[2003, 15M]
17) Show that every finite integral domain is a field.
[2002, 10M]
18) Let \(F\) be a field with \(q\) elements. Let \(E\) be a finite extension of degree \(n\) over \(F\). Show that \(E\) has \(q^{n}\) elements.
[2002, 10M]
19) Show that polynomial \(25 x^{4}+9 x^{3}+3 x+3\) is irreducible over the field of rational numbers.
[2002, 12M]
20) Prove that the polynomial \(1+ x+ x^2 + \cdots x^{p-1}\), where \(p\) is a prime number, is irreducible over the field of rational numbers.
[2001, 12M]
21) Let \(K\) be a field and \(G\) be a finite subgroup of the multipe group of non-zero elements of \(K\). Show that \(G\) is a cyclic group.
[2001, 12M]
22) Prove that every finite extension of a field is an algebraic extension. Give an example to show that the converse is not true.
[2001, 20M]