7.12G: Plasmids as Cloning Vectors
Plasmids can be used as cloning vectors, allowing the insertion of exogenous DNA into a bacterial target.
LEARNING OBJECTIVES
Illustrate how plasmids can be used as cloning vectors
Key Takeaways
Vectors
In molecular biology, a vector is a DNA molecule used as a vehicle to transfer foreign genetic material into another cell. The four major types of vectors are plasmids, viral vectors, cosmids, and artificial chromosomes. All engineered vectors have an origin of replication, a multi-cloning site, and a selectable marker. The vector itself is generally a DNA sequence that consists of an insert (transgene) and a larger sequence, which serves as the “backbone” of the vector. The purpose of a vector that transfers genetic information to another cell is typically to isolate, multiply, or express the insert in the target cell. Vectors called expression vectors (expression constructs) express the transgene in the target cell, and they generally have a promoter sequence that drives expression of the transgene.
Plasmids
Plasmids are double-stranded, generally circular DNA sequences capable of automatically replicating in a host cell. Plasmid vectors minimally consist of the transgene insert and an origin of replication, which allows for semi-independent replication of the plasmid in the host. Modern plasmids generally have many more features, notably a “multiple cloning site”—with nucleotide overhangs for insertion of an insert—and multiple restriction enzyme consensus sites on either side of the insert. Plasmids may be conjugative/transmissible or non-conjugative. Conjugative plasmids mediate DNA transfer through conjugation and therefore spread rapidly among the bacterial cells of a population. Nonconjugative plasmids do not mediate DNA through conjugation.
Transcription
Transcription is a necessary component in all vectors. The purpose of a vector is to multiply the insert, although expression vectors also drive the translation of the multiplied insert. Even stable expression is determined by stable transcription, which depends on promoters in the vector. However, expression vectors have a two expression patterns: constitutive (consistent expression) or inducible (expression only under certain conditions or chemicals). Expression is based on different promoter activities, not post-transcriptional activities, meaning these two different types of expression vectors depend on different types of promoters. Expression vectors require translation of the vector’s insert, thus requiring more components than simpler transcription-only vectors.
Expression vectors require sequences that encode for:
- A polyadenylation tail at the end of the transcribed pre-mRNA: This protects the mRNA from exonucleases and ensures transcriptional and translational termination and stabilizes mRNA production.
- Minimal UTR length: UTRs contain specific characteristics that may impede transcription or translation, so the shortest UTRs are encoded for in optimal expression vectors.
- Kozak sequence: a vector should encode for a Kozak sequence in the mRNA, which assembles the ribosome for translation of the mRNA.
The above conditions are necessary for expression vectors in eukaryotes, not prokaryotes.
Modern vectors may encompass additional features besides the transgene insert and a backbone:
- Promoter: a necessary component for all vectors, used to drive transcription of the vector’s transgene.
- Genetic markers: Genetic markers for viral vectors allow for confirmation that the vector has integrated with the host genomic DNA.
- Antibiotic resistance: Vectors with antibiotic-resistance allow for survival of cells that have taken up the vector in growth media containing antibiotics through antibiotic selection.
- Epitope: A vector containing a sequence for a specific epitope that is incorporated into the expressed protein. Allows for antibody identification of cells expressing the target protein.
- Reporter genes: Some vectors may contain a reporter gene that allow for identification of plasmid that contains inserted DNA sequence. An example is lacZ-α which codes for the N-terminus fragment of β-galactosidase, an enzyme that digests galactose.
- Targeting sequence: Expression vectors may include encoding for a targeting sequence in the finished protein that directs the expressed protein to a specific organelle in the cell or specific location such as the periplasmic space of bacteria.
- Protein purification tags: Some expression vectors include proteins or peptide sequences that allows for easier purification of the expressed protein. Examples include polyhistidine-tag, glutathione-S-transferase, and maltose binding protein. Some of these tags may also allow for increased solubility of the target protein. The target protein is fused to the protein tag, but a protease cleavage site positioned in the polypeptide linker region between the protein and the tag allows the tag to be removed later.
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Key Points
- All engineered vectors have an origin of replication, a multi- cloning site, and a selectable marker.
- Expression vectors (expression constructs) express the transgene in the target cell, and they have a promoter sequence that drives expression of the transgene.
- Transcription is needed for a plasmid to function, without the proper sequences to transcribe parts of a plasmid it will not be expressed or even maintained in host cells.
- Vectors can have many additional sequences that can be used for downstream applications—purification of proteins encoded by the plasmid and expressing proteins targeted to be exported or to a certain compartment of the cell.
Key Terms
- Kozak sequence : a sequence which occurs on eukaryotic mRNA and has the consensus (gcc)gccRccAUGG. The Kozak consensus sequence plays a major role in the initiation of the translation process. The sequence was named after the person who brought it to prominence, Marilyn Kozak.
- transcription : The synthesis of RNA under the direction of DNA.
- polyadenylation : The formation of a polyadenylate, especially that of a nucleic acid


1: Introduction and Review
The Mars Climate Orbiter is prepared for its mission. The laws of physics are the same everywhere, even on Mars, so the probe could be designed based on the laws of physics as discovered on earth. There is unfortunately another reason why this spacecraft is relevant to the topics of this chapter: it was destroyed attempting to enter Mars' atmosphere because engineers at Lockheed Martin forgot to convert data on engine thrusts from pounds into the metric unit of force (newtons) before giving the information to NASA. Conversions are important!
Contributors and Attributions
- Benjamin Crowell (Fullerton College). Conceptual Physics is copyrighted with a CC-BY-SA license.


17.3: Assignment- Personal Reflection
The previous page in this module, Reflective Learning , suggests two alternate ways of viewing our own progression in comprehension and skill development. The last line in the document presents a task:
Use Pappas’s and Tolisano’s taxonomies of reflection to help you reflect on your learning, growth and
development as a reader, writer, and thinker in this course.
Consider questions like these as you work: Do you feel that one or both of these models speaks to your progress in this course? Where would you have placed yourself on either scale at the beginning of the quarter? Here, at the end of the quarter? How might you use this same process to create writing in future courses or in your personal/professional life?
Assignment Requirements
This reflection should consist of 2–3 paragraphs of developed, edited prose. Make clear reference to one or both of the taxonomies in your reflection.
Want to see a sample of what this submission might look like? View 2 student samples available here (Google Doc link).
Contributors and Attributions
- Composition II. Authored by : Elisabeth Ellington, Ph.D.. Provided by : Chadron State College. Located at : http://www.csc.edu/ . Project : Kaleidoscope Open Course Initiative. License : CC BY: Attribution


6.6: Machines can Evaluate Writing Well
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Author: Les Perelman, Comparative Media Studies/ Writing, Massachusetts Institute of Technology
Across the United States, writing is being evaluated by machines. Consider the situation of Maria, a typical high school student. In a high-stakes test situation that could decide whether she’s admitted to the university of her choice, she’s given a prompt to write about a topic such as whether originality is overrated, or whether our society values certainty over skepticism. For the next 25 minutes, she tries to create a purposeful piece of writing, developing its ideas, shaping its structure, considering its style and voice, choosing appropriate examples, and honing it to suggest to its readers—her evaluators—that she can think and write effectively enough to be admitted to a good college. She even drops in a bit of humor to lighten up the essay.
She writes her essay for people who she imagines are like her teachers—people who can read and form conclusions about her essay from multiple angles, know irony or humor when they see it, can spot an unsupported generalization, or can forgive a minor grammatical error while taking note of a more serious one. But instead of reaching those human readers, her essay is fed into a computer system for evaluation. The machine scans the paper for a handful of simple features, such as length and the percentage of infrequently used words. In a few milliseconds, it spits out a score that seals Maria’s fate.
To testing agencies, machine scoring is irresistibly alluring. Instead of hiring, training, and paying warm-blooded human beings to read and judge tens of thousands of essays, they think that investing in a computer scoring system will save them large amounts of money and time and will generate big profits. They have faith that parents, students, school officials, and the general public will think the machines are better than human readers. After all, computers are so accurate and consistent and reliable, right? Why run the risk that the evaluator reading Maria’s essay is cranky or tired on the day of the evaluation, or is coming down with a cold? Machines offer razor-sharp precision and metallic solidity, never giving in to frustration or exhaustion.
But as we’ll show, although computers are brilliant at many things, they’re really bad at understanding and interpreting writing—even writing produced by fifth-graders—and that fact will not change in the foreseeable future. Understanding why this is true can prepare teachers, parents, students, and taxpayers to push back against testing agencies and politicians who think that people will be placated by the complexity of technology and seduced by the promise that writing can be evaluated cheaply and efficiently, justifying further cuts in educational funding.
Why Machines Make Lousy Humans
First, it’s important to understand that computers are not the enemy. In fact, computers play an important role in research on the language and writing that humans produce. There are some things a computer can do in a couple of seconds that would take a human researcher a lifetime (or two). Scholars of writing are the last people to resist the further development of computers to work with natural language—a term referring to the spoken or written language produced by humans as part of their daily lives.
But when it comes to evaluating writing, computers perform badly. That’s because natural language is extraordinarily complex— far more complex than even the most sophisticated computers can understand.
Let’s consider a few reasons why.
- Computers don’t understand meaning. They can compute the likelihood of two words appearing close to each other, but their judgment is always based on statistical probabilities, not an understanding of word connotations. Think of the verb to serve . We can serve our country, serve in a game of tennis, or serve the president. We also can serve a casserole to you. (A cannibalistic restaurant could even serve presidents for lunch, though the supply would be pretty limited.) Humans can easily differentiate between the realistic and absurd meanings of a simple word like serve; computers can’t.
- A computer can’t differentiate between reasonable and absurd inferences either . In fact, computers are really bad at making any inferences at all. When we speak or write, large amounts of information are left out and inferred by the listener or reader. When we read, “Fred realized he couldn’t pay for his daughter’s tuition. He looked up his uncle’s email address,” the space between the two sentences is filled with information that we infer. Almost all human language works this way. Making inferences requires vast amounts of information and astronomically large networks, connections, and permutations in infinite contexts. Although computers can obviously store and search for massive amounts of data, they don’t know how to put it together to infer. The computer would read the two statements above exactly the same as it would read, “Fred realized he couldn’t pay for his daughter’s tuition. He looked up his pet elephant’s email address.”
- Most computer scoring programs judge logical devel opment and effective organization by the number of sentences or words in a paragraph. If a system is programmed to see one-sentence paragra phs as undeveloped, it will apply this principle to all essays even though one-sentence paragraphs can be used to good effect (as in the sentence at the top of this bullet list). When one of us (Perelman) tried to write the best essay he could, one of the most popular machine graders admonished him that a paragraph was underdeveloped because it had only three sentences. He then expanded the paragraph by inserting completely irrelevant material—the opening line of Alan Ginsberg’s poem “Howl”: “I saw the best minds of my generation destroyed by madness, starving hysterical naked.” The computer then considered the new paragraph to be both adequately developed and coherent.
- Computers get progressively worse at evaluating writ ing as it gets longer (for obvious reasons—there’s more to mess them up). The programmers know this. Although all commercial computer-scoring systems give higher scores to longer essays, paradoxically most limit the length of papers to around 1,000 words, about four typed pages. The Educational Testing Service’s program Criterion, for example, almost always gives high scores to essays of 999 words but will refuse to evaluate an essay containing 1,001 words. However, many college papers are more than 1,000 words.
The first myth to debunk about computer scoring systems is that they can read extended discourse, especially to evaluate students’ writing abilities. They can’t. They don’t understand or interpret anything that’s fed into them. They miss humor and irony, or clever turns of phrase, or any of a dozen aspects of prose that we try to teach students. They can’t discern purposeful stylistic decisions. They think gibberish is acceptable, and they mark perfectly reasonable prose that violates some simplistic criterion such as the number of words in a paragraph as unacceptable. They always interpret some aspect of writing the same way, without considering the writer’s intentions and context. They can’t make inferences between the lines of text. The complexity of human language simply baffles them—or, more accurately, goes right over their semiconductors. Writing experts have exposed these and other limitations of machine scoring using both coherent and incoherent essays. The computers can’t tell the difference.
In one experiment, researchers at MIT created the Basic Automated Bullshit Essay Language Generator (BABEL), which produces gibberish essays. When they submitted essays produced by BABEL to a system that scores tens of thousands of student test essays, including the Graduate Record Examination, the computer awarded the gibberish essays the highest possible score. Here is an excerpt of a Babel Generated Essay that received the highest score (6) from ETS’s e-rater, along with the canned comments from ETS’s GRE preparation website.
Careers with corroboration has not, and in all likelihood never will be compassionate, gratuitous, and disciplinary. Mankind will always proclaim noesis; many for a trope but a few on executioner. A quantity of vocation lies in the study of reality as well as the area of semantics. Why is imaginativeness so pulverous to happenstance? The reply to this query is that knowledge is vehemently and boisterously contemporary.
The score: 6
In addressing the specific task directions, a 6 response presents a cogent, well-articulated analysis of the issue and conveys meaning skillfully. A typical response in this category:
- articulates a clear and insightful position on the issue in accordance with the assigned task
- develops the position fully with compelling reasons and/or persuasive examples
- sustains a well-focused, well-organized analysis, connecting ideas logically
- conveys ideas fluently and precisely, using effective vocabulary and sentence variety
demonstrates superior facility with the conventions of standard written English (i.e., grammar, usage, and mechanics) but may have minor errors.
Obviously, the Babel gibberish essay does none of these things. So why, with all these limitations, has computer essay scoring even seen the light of day? We’ve pointed to the economic reasons and the desire for profit. But there’s another reason, and it’s about humans, not computers.
Why Humans Make Lousy Machines
When we look at how humans read and evaluate students’ test essays, we find an interesting paradox. For years, groups of readers have been trained— normed and calibrated —to read thousands of essays in the most consistent and accurate way possible. This is because when we allow people to read writing normally, they see it subjectively, through the lens of their experiences (think of a book club discussion). If a testing agency allowed this—if it couldn’t guarantee consistently of evaluation—it would be instantly sued. Through a long process, readers can often develop consensus on how to evaluate many aspects of papers, but such a process takes more time and money than the testing organizations are willing to spend. Instead, their training process turns humans into machines so that they will look for exactly the same features in exactly the same way, as quickly as possible. They’re told to ignore facts because they can’t verify everything they read. They’re constrained to see the essays only through the lens of what the evaluators think is important. They want to read beyond the lines of the assessment criteria, but they can’t. Because humans are required to read 20–30 essays per hour, they end up evaluating essays using the same simple features used by the machines.
In reading high-stakes, one-shot essay tests, then, both machines and humans make lousy evaluators when we reduce their reading process to a few limited features. Machines do this because they can’t do anything else. Humans do this because they’re trained to ignore everything else they might see and interpret in an essay, including even how factual its assertions are, in order to score only those things that the test makers deem significant and, more importantly, can be scored very quickly (slow, thoughtful reading costs money).
To take a (not so extreme) case, imagine that we assume good writing can be measured entirely by the number of grammatical and punctuation mistakes in a text.  A computer can be similarly trained to recognize a lot of mistakes, even while missing some and flagging false positives. But both evaluators, human and computer, miss the point. Writing is far more complex than a missing comma. The testing agencies that fail to understand fully what writing is and how the ability to produce it is best measured are at fault.
Taking the Machine Out of Writing Assessment
When it comes to testing and evaluating our kids’ writing, machines alone aren’t really the problem. It’s what we’re telling the machines to do. And that’s very similar to what we ask human evaluators to do. What, then, is the solution?
First, we need to stop testing our kids’ writing to death. Computer scientists (who are not writing specialists) were attracted to the possibility of machine scoring precisely because the regressive kind of human scoring they were presented with looked so simple and replicable. We must start by critiquing the testing machine writ large—the multibillion-dollar industry that preys on school districts, misinformed politicians, naïve parents, and exploitable kids under the guise of providing assessment data designed to improve education. Nothing is improved by relentless testing, especially of the kind that reduces writing to the equivalent of running on a hamster wheel. No standardized writing test is purposeful, motivating, or engaging, and it almost never gives the writer any response other than a number.
If the methods of this testing and evaluation are misguided, what happens with the results can be deplorable. Because of relentless and unfounded accountability coming from politicians and government officials who often know next to nothing about how education really works, schools must demonstrate their success through standardized tests. Teachers’ pay raises or even their jobs are linked to their students’ scores on these tests, and entire schools can be defunded or closed if they fall too far below a norm, even though they may be located in an area of urban blight and populated by kids who, through no fault of their own, do not have advantages that support their early literacy development. So what happens? The teachers, fearful of the consequences of poor test scores, begin narrowing everything they do in anticipation of the standardized tests. This process can bankrupt kids’ education by denying them richer learning experiences unrelated to the narrow parameters of the tests. Worse, it bankrupts teachers’ creativity and freedom to apply skills and strategies they’ve learned as educators to create a meaningful, engaging curriculum—in other words, to teach , in the best sense of the word.
What’s the alternative? It’s not in evaluation, but in support . It’s to get testing off the backs of students and teachers. It’s to help young people to develop their writing abilities in authentic situations that give them time to think and formulate ideas, gather necessary information, structure and draft pieces of writing, and hone them to accomplish meaningful goals, such as to inform or persuade or entertain people who can make sense of what they write. It’s to put far more time into teaching than evaluating. It’s to re-empower teachers to use their best, most creative abilities to nurture students’ writing and give them multiple purposes, contexts, and audiences. It’s to recognize the meaning that writers are conveying and not just simple formal elements of their prose. It’s to recognize that students are at different stages of development and language proficiency and to teach accordingly.
Why are we reducing writing situations to sterile, purposeless tasks designed to yield a few metrics that are poorly related to the meaning of the word “writing?” Test makers and evaluation agencies will say that they aren’t denying learners all the rich, meaningful writing situations they should encounter, but that their tests are a convenient, simple, cheap way to measure what they can do. But they’re not. More authentic kinds of evaluation, such as student portfolios carefully read by teachers, are much better and more humane methods because they focus as much on the development of ability as they do on its measurement. And if computers can’t read a 1,000-word test essay, they won’t even begin to know what to do with a portfolio.
Further Reading
For more about the problems with machine evaluation of writing, see Ellis B. Page’s prescient essay from 1966, “The Imminence of Grading Essays by Computer,” Patricia Freitag Ericsson and Richard Haswell’s edited collection Machine Scoring of Student Essays: Truth and Consequences, Les Perelman’s “When ‘the State of the Art’ is Counting Words,” Doug Hesse’s “Who Owns Writing?,” as well as two pieces by Anne Herrington and Charles Moran: “What Happens when Machines Read our Students’ Writing?” and “When Writing to a Machine is Not Writing at All.” For a major professional organization’s stance on machine evaluation of writing, see the National Council of Teachers of English’s position statement, “Machine Scoring Fails the Test.” For more about standardized testing and its problems, see Chris Anson’s “Closed Systems and Standardized Writing Tests,” Todd Farley’s Making the Grades: My Misadventures in the Standardized Testing Industry , and a piece in Slate by Matthew J.X. Malady titled “We are Teaching High School Students to Write Terribly: The Many Problems of the SAT’s Essay Section.”
Keywords
essay grading, high-stakes writing tests, machine scoring, standardized tests, writing assessment
Author Bios
Chris Anson is Distinguished University Professor and director of the Campus Writing and Speaking Program at North Carolina State University, where he works with faculty across the curriculum to improve the way that writing is integrated into all disciplines. For almost four decades, he has studied, taught, and written about writing and learning to write, especially at the high school and college levels. He is past chair of the Conference on College Composition and Communication and past president of the Council of Writing Program Administrators. He has studied and written about writing and computer technology and is a strong advocate of increased attention to digital modes of communication in instruction, but his research does not support the use of computers to score the evaluation of high-stakes writing tests.
Les Perelman recently retired as director of Writing Across the Curriculum in the department of Comparative Media Studies/ Writing at the Massachusetts Institute of Technology, where he has also served as an associate dean in the Office of the Dean of Undergraduate Education. He is currently a research affiliate at MIT. He is a member of the executive committee of the Conference on College Composition and Communication and co-chairs that organization’s Committee on Assessment. Under a grant from Microsoft, Dr. Perelman developed an online evaluation system for writing that allows student access to readings and time to plan, draft, and revise essays for a variety of assessment contexts. Perelman has become a well-known critic of certain standardized writing tests and especially the use of computers to evaluate writing.


4: Ethers and Epoxides; Thiols and Sulfides
We shall begin in a very traditional manner, with a discussion of the nomenclature of ethers. We will then describe how ethers may be prepared in the laboratory, and discuss the relative inertness of these compounds. A discussion of the chemistry of cyclic ethers follows, with particular emphasis on the preparation and reactions of epoxides (cyclic ethers containing a three-membered ring). We will then introduce crown ethers—compounds that consist of large rings containing several oxygen atoms and the spectroscopic properties of ethers. The unit will close with a description of the chemistry of thiols and sulfides, the sulfur-containing analogues of alcohols and ethers.
-
- 4.5: Reactions of Ethers - Claisen Rearrangement
- The Claisen rearrangement is a key organic reaction that involves the thermal rearrangement of allyl vinyl ethers to form β-aryl allyl ethers. It's a valuable tool in organic synthesis, enabling the creation of complex molecular structures with high selectivity and efficiency.
-
- 4.6: Cyclic Ethers - Epoxides
- Cyclic ethers, specifically epoxides, are organic compounds with a ring structure containing an oxygen atom. Epoxides are characterized by a highly strained, reactive three-membered ring. They find applications in organic synthesis, polymerization, asymmetric synthesis, and biology. Despite their utility, they require careful handling due to their reactivity and potential health hazards.
-
- 4.8: Crown Ethers
- Crown ethers are a fascinating class of chemical compounds known for their unique structure and remarkable properties. These molecules consist of a ring-shaped arrangement of oxygen atoms, typically with carbon atoms forming the backbone. The name "crown" comes from their crown-like shape when viewed in three dimensions.
- fulfill all of the detailed objectives listed under each individual section.
- design a multi-step synthesis using one or more of the reactions introduced in this chapter, along with any number of the reactions you have studied to date.
- solve “road-map” problems that may require a knowledge of the chemistry of ethers, epoxides, thiols and sulfides, in addition to any of the material you have studied up to this point in organic chemistry.
- define, and use in context, the key terms introduced in this chapter.


23.8: Sage Exercises
In the analysis of Example \(23.25\) with Sage, two subgroups of order \(2\) and one subgroup of order \(4\) were not analyzed. Determine the fixed fields of these three subgroups.
Build the splitting field of \(p(x)=x^3-6x^2+12x-10\) and then determine the Galois group of \(p(x)\) as a concrete group of explicit permutations. Build the lattice of subgroups of the Galois group, again using the same explicit permutations. Using the Fundamental Theorem of Galois Theory, construct the subfields of the splitting field. Include your supporting documentation in your submitted Sage worksheet. Also, submit a written component of this assignment containing a complete layout of the subgroups and subfields, written entirely with mathematical notation and with no Sage commands, designed to illustrate the correspondence between the two. All you need here is the graphical layout, suitably labeled — the Sage worksheet will substantiate your work.
The polynomial \(x^5-x-1\) has all of the symmetric group \(S_5\) as its Galois group. Because \(S_5\) is not solvable, we know this polynomial to be an example of a quintic polynomial that is not solvable by radicals. Unfortunately, asking Sage to compute this Galois group takes far too long. So this exercise will simulate that experience with a slightly smaller example.
Consider the polynomial \(p(x)=x^4+x+1\text{.}\)
- Build the splitting field of \(p(x)\) one root at a time. Create an extension, factor there, discard linear factors, use the remaining irreducible factor to extend once more. Repeat until \(p(x)\) factors completely. Be sure to do a final extension via just a linear factor. This is a little silly, and Sage will seem to ignore your final generator (so you will want to setermine what it is equivalent to in terms of the previous gfenerators). Directions below depend on taking this extra step.
- Factor the original polynomial over the final extension field in the tower. What is boring about this factorization in comparison to some other examples we have done?
- Construct the full tower as an absolute field over \({\mathbb Q}\text{.}\) From the degree of this extension and the degree of the original polynomial, infer the Galois group of the polynomial.
-
Using the mappings that allow you to translate between the tower and the absolute field (obtained from the
.structure()
method), choose one of the roots (any one) and express it in terms of the single generator of the absolute field. Then reverse the procedure and express the single generator of the absolute field in terms of the roots in the tower. - Compute the group of automorphisms of the absolute field (but don't display the whole group in what you submit). Take all four roots (including your silly one from the last step of the tower construction) and apply each field automorphism to the four roots (creating the guaranteed permutations of the roots). Comment on what you see.
- There is one nontrivial automorphism that has an especially simple form (it is the second one for me) when applied to the generator of the absolute field. What does this automorphism do to the roots of \(p(x)\text{?}\)
- Consider the extension of \({\mathbb Q}\) formed by adjoining just one of the roots. This is a subfield of the splitting field of the polynomial, so is the fixed field of a subgroup of the Galois group. Give a simple description of the corresponding subgroup using language we typically only apply to permutation groups.
Return to the splitting field of the quintic discussed in the introduction to the previous problem (\(x^5-x-1\)). Create the first two intermediate fields by adjoining two roots (one at a time). But instead of factoring at each step to get a new irreducible polynomial,
divide
by the linear factor you
know
is a factor. In general, the quotient might factor further, but in this exercise presume it does not. In other words, act as if your quotient by the linear factor is irreducible. If it is not, then the
NumberField()
command should complain (which it will not).
After adjoining two roots, create the extension producing a third root, and do the division. You should now have a quadratic factor. Assuming the quadratic is irreducible (it is) argue that you have enough evidence to establish the order of the Galois group, and hence can determine exactly which group it is.
You can try to use this quadratic factor to create one more step in the extensions, and you will arrive at the splitting field, as can be seen with logic or division. However, this could take a long time to complete (save your work beforehand!). You can try passing the
check=False
argument to the
NumberField()
command — this will bypass checking irreducibility.
Create the finite field of order \(3^6\text{,}\) letting Sage supply the default polynomial for its construction. The polynomial \(x^6+x^2+2x+1\) is irreducible over this finite field. Check that this polynomial splits in the finite field, and then use the
.roots()
method to collect the roots of the polynomial. Get the group of automorphisms of the field with the
End()
command.
You now have all of the pieces to associate each field automorphism with a permutation of the roots. From this, identify the Galois group and all of its subgroups. For each subgroup, determine the fixed field. You might find the roots easier to work with if you use the
.log()
method to identify them as powers of the field's multiplicative generator.
Your Galois group in this example will be abelian. So every subgroup is normal, and hence any extension is also normal. Can you extend this example by choosing a nontrivial intermediate field with a nontrivial irreducible polynomial that has all of its roots in the intermediate field and a nontrivial irreducible polynomial with none of its roots in the intermediate field?
Your results here are “typical” in the sense that the particular field or irreducible polynomial makes little difference in the qualitative nature of the results.
The splitting field for the irreducible polynomial \(p(x)=x^7-7x+3\) has degree 168 (hence this is the order of the Galois group). This polynomial is derived from an “Elkies trinomial curve,” a hyperelliptic curve (below) that produces polynomials with interesting Galois groups:
\[ y^2 = x(81x^5 + 396x^4 + 738x^3 + 660x^2 + 269x + 48) \nonumber \]
For \(p(x)\) the resulting Galois group is \(PSL(2,7)\text{,}\) a simple group. If \(SL(2,7)\) is all \(2\times 2\) matrices over \({\mathbb Z}_7\) with determinant 1, then \(PSL(2,7)\) is the quotient by the subgroup \(\{I_2,-I_2\}\text{.}\) It is the second-smallest non-abelian simple group (after \(A_5\)).
See how far you can get in using Sage to build this splitting field. A degree \(7\) extension will yield one linear factor, and a subsequent degree \(6\) extension will yield two linear factors, leaving a quartic factor. Here is where the computations begin to slow down. If we believe that the splitting field has degree \(168\text{,}\) then we know that adding a root from this degree 4 factor will get us to the splitting field. Creating this extension may be possible computationally, but verifying that the quartic splits into linear factors here seems to be infeasible.
Return to Example \(23.25\), and the complete list of subfields obtainable from the
.subfields()
method applied to the flattened tower. As mentioned, these are technically not subfields, but do have embeddings into the tower. Given two subfields, their respective primitive elements are embedded into the tower, with an image that is a linear combination of powers of the primitive element for the tower.
If one subfield is contained in the other, then the image of the primitive element for the smaller field should be a linear combination of the (appropriate) powers of the image of the primitive element for the larger field. This is a linear algebra computation that should be possible in the tower, relative to the power basis for the whole tower.
Write a procedure to determine if two subfields are related by one being a subset of the other. Then use this procedure to create the lattice of subfields. The eventual goal would be a graphical display of the lattice, using the existing plotting facilities available for lattices, similar to the top half of Figure 23.26. This is a “challenging” exercise, which is code for “it is speculative and has not been tested.”