These are notes to pp. 1-11 and 21-49 of Artin’s classic—”the last corrected printing of the 1944 second revised edition,” republished by Dover in 1998—to assist the study of his account of Galois theory as far as the Fundamental Theorem. In some places I have followed Artin’s revised German edition (Teubner, 1959), which seems not to have been translated into English. I hope that any mistakes that may be noticed will be brought to my attention.

Format

page or paragraph/line

para = paragraph

–*n* = *n* from bottom

para *n* = the *n*th paragraph that *begins* on the page

1 para 2/-2 is easily seen to be

4 para 2/2 called (**linearly**) **dependent** if

4 para 2/-1 called (**linearly**) **independent**.

8/-7 Theorem 2

8/-4 last

22/-2 [To agree with later pages, a better notation is:]

23/1 [With the “better notation” the condition is:]

23 para 2/-3 less than that of

25 para 1/4 there are non-zero polynomials with this property.

25 para 1/5 non-zero polynomials

25 para 2/-1 [Add at the end of the para:]

The polynomial is called the **minimal polynomial** of over .

26 para 1/3 [“Finally” should start a new para.]

26 para 2/1 We are about to show that

26 para 2/-3 in consequence

26/-4 polynomials in

27/7, 8 the ordinary product

27/12 exceed in degree

28 para 1/12 [Delete “of” at the end of the line.]

29/3 is a field—an extension field of in which has a root, namely .

29 para 2/4 [The following clarifies the text from the sentence beginning “This mapping” through para 4:]

Let be a mapping of fields. For any in , is the **image** of under . The mapping is a **homomorphism** if for all and in ,

(1) and

(2)

Then implies , and then also implies

(3)

Further, implies that either or

In the former case, for all in the mapping is identically zero.

In the latter case, for all in implies

(4) and

The homomorphism is an **isomorphism** if it is both **one-to-one** (that is, it sends distinct elements of to distinct elements of ) and **onto** (that is, every element of is the image of an element of ). In that case there is defined the **inverse** mapping which takes the image of any element back to that element: This mapping too is an isomorphism, as is easily verified. Fields that admit an isomorphism between them are called **isomorphic**.

An important special case is that in which so that the isomorphism is from to itself, it is then called an **automorphism** of Every field admits the **identity** automorphism, which takes each element to itself.

For the general homomorphism let denote the image of under that is, the set of all images of elements of If is onto,

*Lemma h. Let be a non-zero homomorphism of fields. Then is a subfield of and defines an isomorphism of fields *

Proof. If and are distinct elements of hence by (4) above, and (1) and (3), so and are distinct elements of Thus is one-to-one, and it is obviously onto That is a subfield of follows from (1)-(4) above, which show that it is closed under the arithmetic operations in

The mapping that takes the element of to the element of is a non-zero homomorphism whose image is By Lemma h, is a field isomorphic under that mapping to

29/-2 a non-constant polynomial

30 para 2/-1 [Add to the end of the sentence:] under which is the image of

30 para 3/1 [Replace the proof by the following (after the revised German edition):]

Proof. Each element of has the form where is a polynomial in of degree less than that of Map the element to the element where is the polynomial in that corresponds to This mapping is obviously an extension of the given mapping and maps onto By Lemma h, we have only to verify that the extended mapping is a homomorphism. Clearly it sends the sum of two elements to the sum of their images. Likewise for the product, because means that is the remainder of upon division by i.e., there is a polynomial such that (p. 26). Passing to the images of the polynomials gives and putting we have

30 para 6/1 since (as we are about to show) it is

31/3 a field in

31/4 the field within generated

31/5 splitting field

31 para 3/1 [Label the first sentence:] (*)

31 para 3/-1 [Add at the end of the para:] This proves (*).

32 para 1/9 corresponding

32 para 2/1 [Insert at the beginning of the para:]

Applying (*), p. 31, let in be a root of in a root of

34/4 If (supposing ) we

34 para 1/4

34 para 1/-1 [Add at the end of the para:]

It is easily seen that maps the identity element of to and maps the inverse of any element to the multiplicative inverse of its image:

36 para 3/3

36 para 3/4-5 [Replace what follows “we have” to the end of the sentence by:]

37/10 [Insert a comma as shown:] and

37/-7 [After the equation insert a comma, followed by:] valid for every

37/-4 of a field

38/5-6 [Replace what follows “we shall” to the end of the sentence by the following (although it is rendered unnecessary by the definition of inverse already given at 29 para 2/4):]

denote by the **inverse** of (the mapping of into i.e. ).

38/-1 and call the

38 para 3 [After this para insert the following:]

We now introduce a useful notion, which will soon have application.

If is any finite group, it is easy to see that each of the following two sets of elements is the same as (i) where is any given element of (ii) Now let be a group of automorphisms of a field and let be the fixed field of For any element in the sum

of the images of under is called the **trace** of In view of (i), for each hence Thus the trace is fixed under so it lies in Further, because the are independent (Theorem 12, Corollary), there exists such that the trace mapping is not identically zero. Finally, we note that since in view of (ii) the trace can also be written

39/1 fixed field

39/6 seen. Indeed, where denotes the trace. Hence

39 para 2/1 from the Corollary to Theorem 13

39 para 2/-1 [The second term of the equation should have not ]

39/-4 fixed field

39/-3 The Corollary to Theorem 13

40/6 succeed

40 para 1/6 over

41/1 polynomials in

41/2 polynomials in

42 para 2/-2 [For a neater continuation of the proof (after the revised German edition), replace from “We note” to the end of the proof (at 43/13) by the following:]

By renumbering the if necessary, we can assume that We can further arrange it that the trace For if let be such that (p. 38), and multiply the equations by thus making the new Now for each we apply to the th equation, obtaining

Adding these equations gives

because for each (p. 38). Since each is in and this contradicts the assumption that the are linearly independent over

44 para 4/1 in that are not in

44 para 5/4 [At the end of the line:] deg

44/-2 [Delete “a” at the end of the line.]

45/-3 [The text from “We first prove” to the end of the proof of the lemma (at 46 para 1/-1) is here restated. (The lemma deserves to be a theorem, as it is in the revised German edition mostly followed here.)]

A lemma is needed, but we first make a general observation. Let be any extension of a field an automorphism of that leaves fixed, and a polynomial in

(*) *If in is a root of is also a root of *

For

*Lemma. Let be a normal extension of with automorphism group Then is a separable extension of More precisely: if is any element of and are the distinct images of under the automorphisms of then*

*is an irreducible polynomial in with root thus is the minimal polynomial of over *

Proof. Let the automorphisms of be denoted so that are the distinct elements among the is one of them. As we remarked in the discussion of the trace (p. 38), for given the set of the for all is the same as hence application of any to the set of elements permutes its members. Then application of any to permutes its factors, and therefore does not change its coefficients. Since the only elements of which are left fixed by all of the are those of is in and it is obviously separable. By (*) above, any polynomial in having as a root has all of the as roots; consequently it is divisible by This means that is the minimal polynomial of Thus is a separable extension of

46 para 2/2-3 Let be the minimal polynomial of

46 para 4 [See the outline of the proof of Theorem 16, below.]

47/-4 [Delete “other”.]

48 para 1/-2 of which are the identity on is equal

49/2

49/3 of if and only if

________________________________

Outline of the proof of the Fundamental Theorem

The sections P1, P2, … below correspond to the eight paragraphs of the proof.

Abbreviations

the set of

# the number of elements in

is normal is a normal extension of

iso isomorphism

auto automorphism

isos of isomorphisms of that leave fixed

autos of automorphisms of that leave fixed

iff if and only if

restricted to

P1. Part (1), proved here, establishes an iso of sets {subgroups of } {subfields of } under which Here = the fixed field of = the group of autos of

P2. The **order** of a finite group is the number of its elements; the **index** of a subgroup of a finite group is the number of its left cosets. Part (3), proved here, says: the order of = and the index of =

P3. This paragraph and the next establish an iso of sets {left cosets of } {isos of } under which where is any element of Note that need not be an *auto* of it is an *iso* from to some subfield of which may be itself or another one. In this paragraph it is shown that: (i) the map from cosets to isos is well-defined—i.e., if then and (ii) the map is one-to-one. (i) is stated differently (“The elements of in any one coset of map in the same way”); to prove it in the form just given, observe that implies for some in hence if is in

P4. This paragraph shows that the map of P3 from cosets to isos is onto, hence is an iso of sets. Then by P2 = #{left cosets of } = #{isos of }.

P5. It is claimed that Indeed, is in iff for all in i.e. which means that is in i.e. is in

P6. This paragraph proves an independent proposition, which we may state as:

*Corollary 3 to Theorem 14. is normal iff is finite and = #{autos of }.*

P7. is normal

iff = #{autos of } (by P6 applied with since is finite)

iff #{isos of } = #{autos of } (since by P4 = #{isos of })

iff {isos of } = {autos of } —i.e., for all —

iff for all i.e. (by P5), i.e. i.e.

is a normal subgroup of

P8. If and are normal, the iso of P3 can be written {autos of }, and this is an iso of groups, since the multiplications correspond: Thus part (2) is established.

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