Daily Prep 11B -- MTH 350-01

--- tags: mth350, dailyprep --- # Daily Prep 11B -- MTH 350-01 ## Overview Last time we looked at *subrings* of a ring, which a ring $S$ that "lives inside" a larger ring $R$. In Module 11B we look at this from a different point of view: Given a ring $R$ that may or may not have properties we want (like the ability to find a multiplicative inverse for an element), can we embed it as a subring inside a larger ring that *does* have those properties? Those larger rings are called **extensions** of the original ring, and we're particularly interested in taking a ring $R$ and finding an extension that is a field. That's called a **field extension** and it's a fundamental construction for algebra and other areas of mathematics. We'll also take a look at another means of creating a new ring from existing rings, namely the **direct sum**. ## Learning objectives **Basic Learning Objectives:** *Before* our class meeting, use the Resources listed below to learn all of the following. You should be reasonably fluent with all of these tasks prior to our meeting; we will field questions on these, but they will not be retaught. + State the definition of a *subfield* of a field; an *extension* of a field; the *field extension generated by* a subset of a field; and a *simple extension* of a field. + Use the definitions to decide wither a subset of a field is a subfield and to list the elements of a field extension. + State the definition of the *direct sum* of two rings. **Advanced Learning Objectives:** *During and after* our class meeting, we will work on learning the following. Fluency with these is not required prior to class. + Explain line-by-line the proofs of Theorem 9.10 and 9.15. + Given two finite rings, list the elements of, and make operation tables for, their direct sum. ## Resources for learning **Reading:** Read through Investigation 9, starting at the section "Subfields and Field Extensions" and continuing to the section "Concluding Activities". There are several dense proofs here, which you can just skim for now. ## Exercises The exercises for this Daily Prep are found on the Google Form: https://docs.google.com/forms/d/e/1FAIpQLSfsMdN4jMzGAhnmM31wCvmQG6JOE8Bjwwumm6ATCHwl3Fsz3w/viewform ## Submission and grading To submit your work, simply submit the Google Form. You will receive a receipt via email to confirm your submission. (Look in your spam folders if you do not see the receipt.) A **Pass** mark is given if the Daily Prep is turned in before its deadline and if each item on the Daily Prep has a response that represents a good faith effort to be right. **Mistakes are not penalized**. A **No Pass** is given if an item is left blank (even accidentally), has an answer but it shows insufficient effort (including responses like "I don't know"), or if the Daily Prep is late.