Hello,(adsbygoogle = window.adsbygoogle || []).push({});

I was doing self studying abstract algebra from the online lecture notes posted by Robert Ash and I hit against the following theorem. I am posting it in the topology section because without a geometric/topological meaning to the concept I am never able to understand the topic and that is the reason during my undergraduate days 8 years ago I did not pass in algebra at all.

Now for the theorem: Let E/F be a finite separable extension of degree n and let sigma be an embedding of F in C (where C is the algebraic closure of E). Then sigma extends to exactly n embeddings of E in C. In other words there are exactly n embeddings of tau in C such that restriction of tau to F coincides with sigma. In particular if sigma is the identity function on F then there are exactly n F-monomorphism of E into C.

The above is reproduced in ditto from Thereom 3.5.2 of R.Ash lecture notes posted on his website.

Now my question.

Let me think of E as R^3 and F as R. Now surely R^3 is the closure of R. (The reason why I have taken R^3 is simply because I am successful in proving the theorem in 2D, using x^2 +1 = 0 and things like that. Now when I tried using x^3 +2 = 0, I understood that I would be getting 6 dimensions of the separable field. )

My hunch would be automorphism could be thought of as vector rotations and hence I thought of coming out with this simple example. Surely having T(a) = a + 1 is an isomorphism but not an automorphism which fixes a.

So I thought of taking the following isomorphisms of a vector .

(x, y, z) ---> (x, y, z) (identity)

(x, y, z) ---> (y, x, z)

(x, y, z) ---> (z, y, x)

(x, y, z) ---> (x, z, y)

(x, y, z) ---> (z, x, y)

(x, y, z) ---> (y, z, x)

The above are some kind of transformations I thought which are possible.

If the theorem above is true then at least 3 of the above isomorphisms should not be an automorphism. I am really not sure which 3 of them should not be and why not. I have tried a lot to think but have been unsuccessful.

The theorem says that the transformation when restricted to F then it should fix F. Now I could think that since the extension of R^3 is over R hence x should be fixed but I only get 2 automorphisms. The identity and (x, y, z) ---> (x, z, y). Not sure why I am not getting the third one, because the theorem says that E/F is of degree n ( which is 3) and so it should be possible to get solutions of all the polynomials in R^3. So there is sth wrong in my understanding or the way that I am thinking of automorphisms as rotations and missing sth.

I thought about whether a splitting field can have 3 dimensions and found that it could. Consider x^3 + 2 = 0 over F. Now it has got 3 roots. 2 complex one real but even the real does not fall in F. So the extension field is (modelling on R^3 ), x axis is F, y axis is (2)^1/3 , z complex root.

Rgds

SM

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Misunderstanding of isomorphism and automorphism

**Physics Forums | Science Articles, Homework Help, Discussion**