Determine Aut(K/Q): Calculating Automorphism Group of K=Q(21/4)

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In summary, Aut(K/Q) is the group of automorphisms of K that fix Q. There are exactly four of these automorphisms, which are determined by their action on 4√2: 4√2, -4√2, (4√2)i, and -(4√2)i. These correspond to the four homomorphisms from K to C, and all other homomorphisms from K to C are not automorphisms.
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Homework Statement



Let K = Q(21/4)

Determine the automorphism group Aut(K/Q)


Homework Equations



An automorphism is an isomorphism from a Field to itself

Aut(K/Q) is the group of Automorphisms from k/Q to K/Q

Definition: A K-Homomorphism from L/K to L'/K is a homomorphism L---> L' that is the identity on K

The Attempt at a Solution



I am completely at a loss really. I have calculated there are four homomorphisms from K to C and think from there if I know how many are K-homomorphisms then that'll be the number of automorphisms, because a homomorphism from a field to itself is an automorphism (Please correct me if I'm wrong on this). Then that'll give me the set of Automorphisms.

My problem is that I don't know how to go from the number of homomorphisms to the actual homomorphisms. I think it has a relation to the roots of 2(1 /4) in C (which I have calculated to be 2(1 /4), - 2(1 /4), i*2(1 /4), -i2(1 /4) )

Please help, this lack of understanding is preventing me from moving forward with other questions and my notes from lectures completely gloss over how to do this.
 
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one problem that I see is that not every homomorphism is an automorphism. Consider the identity mapping. this is clearly a homomorphism but it is not bijective.
Also you said there are four homomorphisms from K to C. But what is C? is C just Aut(K/Q)?
Also I just want to clarify something: Is this what you mean by K:

[tex] K= \{ a+b \sqrt[4]{2} :a,b \in \mathbb{Q} \} [/tex]
 
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  • #3
no C is not Aut(K/Q).

Aut(K/Q) is the group of all automorphisms of K that fix Q.

note K is NOT {a+b(4√2)}, this set is not even closed under multiplication, K is more precisely the set:

{a + b(4√2) + c√2 + d(4√8)}

which as a vector space is isomorphic to Q4.

what IS clear, is that any element of Aut(K/Q) sends 4√2 to another 4th root of two, and that any such automorphism is completely determined by the image of 4√2.

there are exactly 4 of these in C:

4√2, -4√2, (4√2)i and -(4√2)i.

if we agree to set α = 4√2, of the maps:

α→α
α→-α
α→iα
α→-iα

how many of these are in Aut(K/Q)?

(the complex numbers play only an indrect role, here, as a field contaning the algebraic closure of Q(α). we could have just used this field (call it E) instead, but it doesn't lend itself to a nice easy description like C does. the important fact is that there is SOME subfield of C that is the algebraic closure of Q(α), as Q(α) is a subfield of R, and C is the algebraic closure of R (a fact called the fundamental theorem of algebra). note we don't even need all of C, just A, where A is the set of all (real) algebraic numbers).
 

What is Aut(K/Q)?

Aut(K/Q) refers to the automorphism group of the field extension K/Q, which is the group of all automorphisms (bijective homomorphisms) from K to itself that fix all elements of the base field Q.

How is Aut(K/Q) calculated?

To calculate Aut(K/Q), we need to determine all possible automorphisms of K that fix all elements of Q. This can be done by considering the action of automorphisms on the basis of K over Q, and then using these actions to construct all possible automorphisms.

What is the significance of calculating Aut(K/Q)?

Calculating Aut(K/Q) is important because it allows us to understand the structure of the field extension K/Q. It also helps us identify important properties of K, such as whether it is Galois or not.

What are some key properties of Aut(K/Q)?

Some key properties of Aut(K/Q) include that it is a group under composition of functions, it is a subgroup of the group of all automorphisms of K, and its order is always less than or equal to the degree of the field extension [K:Q].

Are there any tools or techniques to simplify the calculation of Aut(K/Q)?

Yes, there are some tools and techniques that can make the calculation of Aut(K/Q) easier. These include using Galois theory, which provides a way to determine the order of Aut(K/Q), and using properties of the field extension K/Q, such as its degree and Galois group, to narrow down the possible automorphisms.

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