## 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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**Sample text**

If - , and if : is a subset of , , the smallest subfield of , containing both and : is denoted by - ²:³. When : ~ ¸ Á Ã Á ¹ is a finite set, it is customary to write - ² Á Ã 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.