Show that the prime subfield of a field of characteristic p is ringisomorphic to Zp and that the prime subfield of a field of characteristic 0 is ring-isomorphic to Q.

Wierzycaz 2021-02-09 Answered

Show that the prime subfield of a field of characteristic p is ringisomorphic to \(Z_{p}\) and that the prime subfield of a field of characteristic 0 is ring-isomorphic to Q.

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Plainmath recommends

  • Ask your own question for free.
  • Get a detailed answer even on the hardest topics.
  • Ask an expert for a step-by-step guidance to learn to do it yourself.
Ask Question

Expert Answer

Derrick
Answered 2021-02-10 Author has 5902 answers

Let us first prove the following.
First Part: If F is a field of characteristic 0, then the prime subfield of F is isomorphic to Q.
Answer: Let us define a homomorphism

\(\phi:Z\rightarrow F\phi:Z\rightarrow F\) by \(\times \phi (n)=n\times 1F.\)
Clearly, the homomorphism \(\phi\) is injective, because F has characteristic 0. It follows that \(\phi (Z)\) is a subring of F and is isomorphic to Z.

Therefore, F contains a subfield isomorphic to Q. Since, Q has no subfields and so it is prime.

We know that, every field contains a unique prime subfield. Therefore, Q is the unique prime subfield of F.
Second Part: If F is a field of characteristic p,p, then the prime subfield of F is isomorphic to \(Z_{p}\) .
Answer: Let us consider \(\phi :Z\rightarrow F\) as define in First Part. In this case, F has characteristic p, it follows that
Ker \(\phi =pZ\)

Therefore, by the first isomorphism theorem that \(\phi (Z)\) is isomorphic to \(\frac{Z}{pZ}.\)

Clearly \(\frac{Z}{pZ}.\) is a prime field. Since, every field contains a unique prime subfield. It follows that \(\frac{Z}{pZ}\) is the unique prime subfield of F.

Have a similar question?
Ask An Expert
45
 

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Relevant Questions

asked 2021-01-04

Let F be a field and consider the ring of polynominals in two variables over F,F[x,y]. Prove that the functions sending a polyomial f(x,y) to its degree in x, its degree in y, and its total degree (i.e, the highest \(i+j\) where \(\displaystyle{x}^{{i}}{y}^{{i}}\) appears with a nonzero coefficient) all fail o be norm making F[x,y] a Euclidean domain.

asked 2021-02-05

Let \(\mathbb{R}\) sube K be a field extension of degree 2, and prove that \(K \cong \mathbb{C}\). Prove that there is no field extension \(\mathbb{R}\) sube K of degree 3.

asked 2021-02-25

If U is a set, let \(\displaystyle{G}={\left\lbrace{X}{\mid}{X}\subseteq{U}\right\rbrace}\). Show that G is an abelian group under the operation \(\oplus\) defined by \(\displaystyle{X}\oplus{Y}={\left({\frac{{{x}}}{{{y}}}}\right)}\cup{\left({\frac{{{y}}}{{{x}}}}\right)}\)

asked 2021-02-26

Let H be a normal subgroup of a group G, and let \(m = (G : H)\). Show that
\(a^{m} \in H\)
for every \(a \in G\)

asked 2021-01-05

Let a,b be coprime integers. Prove that every integer \(x>ab-a-b\) can be written as \(na+mb\) where n,m are non negative integers. Prove that \(ab-a-b\) cannot be expressed in this form.

asked 2021-02-27

Prove the following.
(1) \(Z \times 5\) is a cyclic group.
(2) \(Z \times 8\) is not a cyclic group.

asked 2021-02-27

In the froup \(\displaystyle{Z}_{{12}}\), find \(|a|, |b|\), and \(|a+b|\)
\(a=5, b=4\)

Plainmath recommends

  • Ask your own question for free.
  • Get a detailed answer even on the hardest topics.
  • Ask an expert for a step-by-step guidance to learn to do it yourself.
Ask Question
...