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单词 QIsThePrimeSubfieldOfAnyFieldOfCharacteristic0ProofThat
释义

Q is the prime subfield of any field of characteristic 0, proof that


The following two propositionsPlanetmathPlanetmath show that can be embedded in any field of characteristic 0, while 𝔽p can be embedded in any field of characteristic p.

Proposition. is the prime subfieldMathworldPlanetmath of any field of characteristic 0.

Proof.

Let F be a field of characteristic 0.  We want to find a one-to-one field homomorphism ϕ:F.  For  mn  with m,n coprimeMathworldPlanetmath, define the mapping ϕ that takes mn into m1Fn1FF.  It is easy to check that ϕ is a well-defined function.  Furthermore, it is elementary to show

  1. 1.

    additivePlanetmathPlanetmath: for p,q, ϕ(p+q)=ϕ(p)+ϕ(q);

  2. 2.

    multiplicative: for p,q, ϕ(pq)=ϕ(p)ϕ(q);

  3. 3.

    ϕ(1)=1F, and

  4. 4.

    ϕ(0)=0F.

This shows that ϕ is a field homomorphism. Finally, if ϕ(p)=0 and p0, then 1=ϕ(1)=ϕ(pp-1)=ϕ(p)ϕ(p-1)=0ϕ(p-1)=0, a contradictionMathworldPlanetmathPlanetmath.∎

Proposition. 𝔽p (/p) is the prime subfield of any field of characteristic p.

Proof.

Let F be a field of characteristic p. The idea again is to find an injective field homomorphism, this time, from 𝔽p into F. Take ϕ to be the function that maps m𝔽p to m1F. It is well-defined, for if m=n in 𝔽p, then p(m-n), meaning (m-n)1F=0, or that m1F=n1F, (showing that one element in 𝔽p does not get “mapped” to more than one element in F). Since the above argumentPlanetmathPlanetmath is reversible, we see that ϕ is one-to-one.

To completePlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath the proof, we next show that ϕ is a field homomorphism. That ϕ(1)=1F and ϕ(0)=0F are clear from the definition of ϕ. Additivity and multiplicativity of ϕ are readily verified, as follows:

  • ϕ(m+n)=(m+n)1F=m1F+n1F=ϕ(m)+ϕ(n);

  • ϕ(mn)=mn1F=mn1F1F=(m1F)(n1F)=ϕ(m)ϕ(n).

This shows that ϕ is a field homomorphism.∎

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更新时间:2025/5/4 15:13:59