MHB Could you explain me about 'relation algebraic property with conjugate'?

[bw0young0math]

Hello everyone. At first, I appreciate your click this page.

I have a book named 'A first Course in Abstract Algebra 7th' by Fraleigh.

I have a question about 'relation algebraic property with conjugate' in automorhisms of fields.
in page415,
this book explains "Let E is algebraic extension of F& a,b∈E. Then a and b have the same algebraic property iff irr(a,F)=irr(b,F)."
What's mean algebraic property in that sentense? If you explain me, I will be happy:)Thanks. Have a nice day:)

 

[caffeinemachine]

Hello everyone. At first, I appreciate your click this page.

I have a book named 'A first Course in Abstract Algebra 7th' by Fraleigh.

I have a question about 'relation algebraic property with conjugate' in automorhisms of fields.
in page415,
this book explains "Let E is algebraic extension of F& a,b∈E. Then a and b have the same algebraic property iff irr(a,F)=irr(b,F)."
What's mean algebraic property in that sentense? If you explain me, I will be happy:)Thanks. Have a nice day:)

I don't have the book. But what Fraleigh is trying to say is the following:

Let $a, b\in K$ be algebraic over $F$ satisfying $irr(a, F)=irr(b, F)$. Then there is a "natural isomorphism" between $F(a)$ and $F(b)$ which is identity on $F$.

The isomorphism is given by $\phi:F(a)\to F(b)$, where $\phi(p(a))=p(b)$ for all $p(x)\in F[x]$.

The fact that $\phi$ is well defined requires $irr(a, F)=irr(b, F)$.

 

[bw0young0math]

I don't have the book. But what Fraleigh is trying to say is the following:

Let $a, b\in K$ be algebraic over $F$ satisfying $irr(a, F)=irr(b, F)$. Then there is a "natural isomorphism" between $F(a)$ and $F(b)$ which is identity on $F$.

The isomorphism is given by $\phi:F(a)\to F(b)$, where $\phi(p(a))=p(b)$ for all $p(x)\in F[x]$.

The fact that $\phi$ is well defined requires $irr(a, F)=irr(b, F)$.

Thanks! I understand it! F(a)and F(b) are isomorphic so we can think that they have the same algebraic constructure and algebraic properties. Thank you:)