Becker and Rosenberg have generalised the notions of orders and the reduced Witt ring of a field to those of signatures and the reduced Witt rings of higher level. An abstract setting for the ordinary reduced Witt ring has been defined by Marshall. This can be generalised to an abstract setting for the reduced Witt rings of higher level, where the objects studied are called Spaces of Signatures. When these come from the original setting they are called realisable.

We examine two operations which allow us to construct new Spaces of Signatures from old ones. Spaces of Signatures built up from the smallest one using these operations are called finite constructable. We show that finite constructable Spaces of Signatures are precisely those that are realisable under a certain finiteness condition. We then apply these results to give an inductive construction of the reduced Witt rings of higher level of these fields. We also obtain results on the possible number of signatures of these fields.

We then study some small Spaces of Signatures and show that they are finite constructable. Thus these small Spaces of Signatures are realisable.

She loved to joke that she got the job because she was interviewed across the road from where she was born. She was promoted to associate professor in 1993 and to full professor in 2006. Vicki was the first woman internally promoted to full professor in the history of the department.

This paper represents a portion of the author's doctoral thesis, written under the supervision of Professor Alex Rosenberg at Cornell University. The author gratefully acknowledges the help and encouragement of Professor Rosenberg.

Powers says she has loved her time at Emory. "Sometimes I walk around the Quad and say to myself, 'I can't believe I work here.'" During an evaluation site visit, one reviewer was tasked with interviewing every member of the department. "At the end of mine, the reviewer said, 'I can't believe it. You're the only person I have talked to who is completely happy,'" Powers says.

At their first meeting, Powers asked not to be called doctor, but by her first name. "She made me feel welcome from day one. Coming from Vietnam, it was such a big adjustment and in graduate school, you're intimidated a bit. She made me feel comfortable sharing or asking questions," Nguyen says. Powers was also able to break down her high-level, complex research in a way that was easy to digest. "She shared her written explanation of her research so that I could model it in my earlier days, and took it upon herself to publish our work when I was adapting to my first job with a busy teaching assignment," Nguyen says. In 2010 Nguyen earned the Marshall Hall, Jr., Award recognising outstanding teaching by a doctoral student in Emory's mathematics department. "Going from someone who was so intimidated to then receiving that award was a lot for me. I couldn't have done that without Vicki," she says. "She's been an inspiration and phenomenal advisor. She stayed in touch after graduation and supported me through my ups and downs."

In her scholarly work, Vicki's initial training and expertise in ordered fields and the algebraic theory of quadratic forms gave way to research in an important area of mathematics known as algebraic geometry. In particular, she is known for her work on positive polynomials, and she also developed a passion for the mathematics of voting systems. She is the author of numerous papers, and the recent Springer book, Certificates of Positivity for Real Polynomials - Theory, Practice and Applications published in 2021. She frequently travelled for collaborations and conferences throughout the USA and Canada, as well as to Europe, especially France and Germany. She also gave numerous workshops including some in Nigeria.

The book under review is a very nice introduction to the central topic of real algebra, namely, algebraic certificates for positive (pos) polynomials on semialgebraic sets K, their applications and their computational aspects. It is aimed mainly at non-experts in the field and as such is accessible to a broad audience with a knowledge of undergraduate linear and abstract algebra. It starts with the birth of real algebra at the end of the 19th century with Hilbert's work on pos polynomials and ends with the state of the art in 2021, when the book was finished. It is divided into 12 chapters with references at the end of each of them, starting with global certificates of positivity (K = ℝn) and then covering special cases (K being a general semialgebraic set, a compact semialgebraic set, having dimension 1 or being a polytope). Two special topics are also certificates for polynomials with coefficients from ℚ and for polynomials with additional properties. The book is very well written with many examples demonstrating the statements of the main results. It mostly omits the proofs, but in each chapter there are at least some of them or sketches so that the reader can get some insight into the intriguing world of real algebra. A brief history of the results in each section is presented with connections to recent research on the topics covered so that an interested reader can pursue the area further on his/her/their own. Most of the chapters also have sections on applications of the results presented. In the reviewer's opinion this is a very nice and concise presentation of the most important pillars of real algebra up to the present time.

Vicki was initially very angry and depressed and terrified to learn of her diagnosis ... But in time, she showed enormous courage and determination. She got in three trips: to Tucson, New York City, and Ireland.

She loved family, music, friends, books, films, gardening, food, and singing. She was a consummate cook and host. She was an avid international traveller for both fun and work, spending much time in Canada, Ireland, Britain, Germany, France, Italy, and Spain, over the years - also visiting Thailand, Myanmar, India, China, and Nigeria. She gave back to the community in many ways, ranging from teaching and mentoring to advocacy and assorted volunteer work. She had a quick mind, a sweet and generous disposition, an irreverent Pythonesque sense of humour, a smile to die for, and a heart of gold.

So many memories of this remarkable woman - we met at my very first conference, when I was a grad student and Vicki a recent PhD. After I finished my degree, Vicki extended to me my first invitation to present at a conference, a kindness that meant so much professionally and personally. We were pregnant with our first-borns at the same time, shared hotel rooms and hosted each other for math meetings, met up with our families including our moms while vacationing in Ireland, held the same position at NSF sequentially... I so valued our connections over the years, even though we saw each other infrequently. And that incredible smile will remain with me forever. Colm, Ann, and Molly, I know she lives on in your hearts, but oh my how her absence in this world leaves an enormous gap. My deepest sympathy on this unimaginable loss.