Field Theory by Steven Roman

By Steven Roman

Meant for graduate classes or for self sufficient examine, this booklet offers the fundamental thought of fields. the 1st half starts with a dialogue of polynomials over a hoop, the department set of rules, irreducibility, box extensions, and embeddings. the second one half is dedicated to Galois thought. The 3rd a part of the publication treats the idea of binomials. The ebook concludes with a bankruptcy on households of binomials - the Kummer concept.

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If -  , and if : is a subset of , , the smallest subfield of , containing both and : is denoted by - ²:³. When : ~ ¸ Á à Á  ¹ is a finite set, it is customary to write - ² Á à Á  ³ for - ²:³. Definition Let -  , . Then 1) , is algebraic over - if every element   , is algebraic over - . Otherwise, , is transcendental over - . 2) , is finitely generated over - if , ~ - ²:³, where : ‹ , is a finite set. 3) , is a simple extension of - if , ~ - ²³, for some   , . In this case,  is called a primitive element of , .

For the converse, if - is a finite field, then so is ,, since it is finite-dimensional over - and so the multiplicative group , i of nonzero elements of , is cyclic. If  generates this group, then , ~ - ²³ is simple. Now suppose that - is an infinite field and there are only finitely many intermediate fields between , and - . Consider the intermediate fields - ² b  ³, for all   - . By hypothesis, - ² b  ³ ~ - ² b   ³ for some  £   - . Hence,  b    - ² b  ³, implying that  ~  ´² b  ³ c ² b   ³µ  - ² b  ³ c and  ~ ² b  ³ c   - ² b  ³ Hence, - ² Á  ³ ‹ - ² b  ³.

7. Let 9 ~ {´jc µ ~ ¸ b  jc “ Á   {¹. Show that this integral domain is not a unique factorization domain by showing that  9 has essentially two different factorizations in 9 . Show also that the irreducible element  is not prime. 8. Let 9 be a pid. Then an ideal ? of 9 is maximal if and only if ? ~ º» where  is irreducible. Also, 9°º» is a field if and only if  is irreducible. 9. Prove that º%» and ºÁ %» are both prime ideals in {´%µ and that º%» is properly contained in ºÁ %». 10.

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