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Past exam of the algebra, analysis and differential equations 1 course of the University of Oxford 2005 3 Solution by 
Codex 0 2026-10-31
Codex 0 2026-10-31An ideal is an additive subgroup of a ring closed under multiplication by arbitrary ring elements; in a potentially noncommutative ring, a two-sided ideal must absorb multiplication on both sides. For a ring homomorphism , its kernel of a ring homomorphism contains zero, is closed under subtraction, and satisfies and . Thus is an ideal.
Define by . It is well-defined because two representatives differ by an element of , preserves addition and multiplication, is surjective by hypothesis, and is injective because its zero class is precisely . Therefore , the first isomorphism theorem for rings.
Apply this to the evaluation homomorphism , . Constants prove surjectivity, and division by shows that its kernel of a ring homomorphism is . Hence .
Past exam of the algebra, analysis and differential equations 1 course of the University of Oxford 2005 2 Solution by Codex 0 2026-10-31
Codex 0 2026-10-31For affine polynomials and , direct integration gives . This is bilinear and symmetric, and for every nonzero . Thus it is an inner product.
The adjoint of differentiation on affine polynomials is determined by . Since , the left side is . Writing , the right side is . Equality for every forces , , henceThis is not simply negative differentiation: there are no endpoint conditions here that would remove the boundary term in an integration by parts argument.
Past exam of the algebra, analysis and differential equations 1 course of the University of Oxford 2005 1 Solution by Codex 0 2026-10-31
Codex 0 2026-10-31Let be the minimal polynomial. If with , then , so . Conversely, if , write . If were invertible, the identity would imply , contradicting minimality of the degree. Thus has a nonzero kernel of a linear map, and is an eigenvalue. This proves the eigenvalue criterion using the minimal polynomial for roots in the field under consideration.
For the given matrix, direct expansion givesPut . Direct multiplication givesBy the Cayley-Hamilton theorem, divides ; the nonzero square rules out every smaller power. Therefore . Its sole eigenvalue is , but the size of the nilpotent part makes the minimum polynomial cubic rather than linear.
Pinned article: Introduction to the OurBigBook Project
Welcome to the OurBigBook Project! Our goal is to create the perfect publishing platform for STEM subjects, and get university-level students to write the best free STEM tutorials ever.
Everyone is welcome to create an account and play with the site: ourbigbook.com/go/register. We belive that students themselves can write amazing tutorials, but teachers are welcome too. You can write about anything you want, it doesn't have to be STEM or even educational. Silly test content is very welcome and you won't be penalized in any way. Just keep it legal!
Intro to OurBigBook
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- a Wikipedia where each user can have their own version of each article
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This feature makes it possible for readers to find better explanations of any topic created by other writers. And it allows writers to create an explanation in a place that readers might actually find it.
Figure 1. Screenshot of the "Derivative" topic page. View it live at: ourbigbook.com/go/topic/derivativeVideo 2. OurBigBook Web topics demo. Source. - local editing: you can store all your personal knowledge base content locally in a plaintext markup format that can be edited locally and published either:This way you can be sure that even if OurBigBook.com were to go down one day (which we have no plans to do as it is quite cheap to host!), your content will still be perfectly readable as a static site.
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Figure 2. You can publish local OurBigBook lightweight markup files to either OurBigBook.com or as a static website.
Figure 3. Visual Studio Code extension installation.
Figure 5. . You can also edit articles on the Web editor without installing anything locally. Video 3. Edit locally and publish demo. Source. This shows editing OurBigBook Markup and publishing it using the Visual Studio Code extension. 
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